**7.3 Traditional/medicinal uses**

In addition to their nutritional benefits, root vegetables also have medicinal value, and their consumption has been used traditionally to treat digestive tract problems, morning sickness, arthritis, high level of cholesterol, etc. They are also good for

healthy eyes and good-looking skin. Garlic posseses digestive, carminative and antirheumatic properties. Since acient times, garlic is used in formulation of Ayurveda curing muscular pain, giddiness, lungs, heating intestinal ulcer, etc. Different phyto-remedies including ginger, turmeric, onion, and garlic have achieved foremost importance in prevention of metabolic syndrome. Ginger is well known for its medicinal properties, ginger can help with alleviating cough, flu, rheumatoid arthritis and travel sickness based on reports. It aids digestion and said to be an inflammation fighter. It is also said to reduce cholesterol and blood pressure level and provides better blood circulation. Ginger is recognized as one of the most important constituents in herbal medicine. Herbal medicine practitioners use ginger as medicine for treating many diseases.


**7. Efficiency of the fetal cardiac scan**

Although the most frequent congenital malformations, CHDs are among the most frequently missed [18, 55]. The efficiency of the cardiac scan is reported with great variation, depending on the scanning protocol, examiner experience, equipment quality and scanning conditions [18, 56–58]. It appears that the use of 4CV alone detects up to 77% of CHD, while adding OTV increases prenatal detection to 83–92% of major abnormalities.


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Yang, L., Tan, C., Hsieh, M-J., Wang, J., Duan, Y., Cieplak, P., Caldwell, J., Kollman, P.A. & Luo, R. (2006). New generation Amber united-atom force field. *J. Phys. Chem. B*, Vol.

Zacharias, N. & Dougherty, D.A. (2002). Cation-π interactions in ligand recognition and catalysis. *Trends Pharm. Sci.,* Vol. 23, No. 6, (June 2002), pp. 281-287. ISSN: 0165-6147 Zgarbová, M., Otyepka, M., Sponer, J., Mládek, A., Banés, P., Cheatham, T.E. III & Jurecka, P. (2011). Refinement of the Cornell et al. nucleic acids force field based on reference quantum chemical calculations of glycosidic torsion profiles. *J. Chem. Theory Comp.*, Vol.

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**Chapter 9**

© 2012 Martínez-Castilla and Rodríguez-Sotres, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is

© 2012 Martínez-Castilla and Rodríguez-Sotres, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

properly cited.

**On the Assessment of Structural Protein**

**Value, Potential and Limitations**

León P. Martínez-Castilla and Rogelio Rodríguez-Sotres

improve the model. The limitations of the protocol are also discussed.

**2. The folding problem is a NP-hard problem involving a degenerate**

As implied by the well-known Levinthal paradox (Levinthal, 1968), a full exploration of the entire conformational space theoretically available to a protein is out of the reach of current computational techniques. Equally unaccessible to nature is the sequence space available to polypeptide chains (Kono & Saven, 2001). Currently, the amount of available protein structures (the PDB) represents a fraction of the known protein amino acid sequences, and if the available sample is grouped in terms of different folds, the diversity in the PDB is even smaller. In addition, protein structure and function can tolerate a significant number of

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/47842

**1. Introduction**

**informational code**

**Models with ROSETTA-Design and HMMer:**

The prediction of the three-dimensional structure of a protein, starting with the amino acid sequence, is still an unsolved issue. However a number of important advancements have been made and some methods offer solutions to this problem, specially when the target sequence has homologues whose structure has been determined. In any case, it is important to evaluate the quality of the prediction, as none of the methods offers assurance of success. The ROSETTA-design-HHMer (Rd.HMM) protocol stands out among the current quality assessment methods, because it offers evidence of the biological appropriateness of the prediction. In addition, Rd.HMM can be used to guide the modeling process towards the improvement of the model's quality. This chapter deals with the principles behind this protocol and gives practical advice on how to use the Rd.HMM to evaluate the quality of a three-dimensional modeled structure of a protein, and how to use the information to


# **New Technologies for the Treatment of Coronary and Structural Heart Diseases**

Alberto Polimeni Edited by

Printed Edition of the Special Issue Published in *Journal of Clinical Medicine*

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**New Technologies for the Treatment of Coronary and Structural Heart Diseases**

• Alberto Polimeni

## **New Technologies for the Treatment of Coronary and Structural Heart Diseases**

## **New Technologies for the Treatment of Coronary and Structural Heart Diseases**

Editor

**Alberto Polimeni**

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*Article* **Biomass and Leaf Nutrition Contents of Selected Grass and Legume Species in High Altitude Rangelands of Kashmir Himalaya Valley (Jammu & Kashmir), India**

**Javed A. Mugloo 1 , Mehraj ud din Khanday 2 , Mehraj ud din Dar 1 , Ishrat Saleem 1 , Hesham F. Alharby 3,4 , Atif A. Bamagoos 3 , Sameera A. Alghamdi 3 , Awatif M. Abdulmajeed 5 , Pankaj Kumar 6 and Sami Abou Fayssal 7,8,\***

**Abstract:** The yield and nutritional profile of grass and legume species in Kashmir Valley's rangelands are scantly reported. The study area in this paper included three types of sites (grazed, protected, and seed-sown) divided into three circles: northern, central, and southern Kashmir. From each circle, three districts and three villages per district were selected. Most sites showed higher aboveground biomass (AGB) compared to belowground biomass (BGB), which showed low to moderate effects on biomass. The comparison between northern, central, and southern Kashmir regions revealed that AGB (86.74, 78.62, and 75.22 t. ha−1 ), BGB (52.04, 51.16, and 50.99 t. ha−1 ), and total biomass yield (138.78, 129.78, and 126.21 t. ha−1 ) were the highest in central Kashmir region, followed by southern and northern Kashmir regions, respectively. More precisely, AGB and total biomass yield recorded the highest values in the protected sites of the central Kashmir region, whereas BGB scored the highest value in the protected sites of southern Kashmir region. The maximum yield (12.5 t. ha−1 ) recorded among prominent grasses was attributed to orchard grass, while the highest crude fiber and crude protein contents (34.2% and 10.4%, respectively), were observed for Agrostis grass. The maximum yield and crude fiber content (25.4 t. ha−1 and 22.7%, respectively), among prominent legumes were recorded for red clover. The highest crude protein content (33.2%) was attributed to white clover. Those findings concluded the successful management of Kashmir rangelands in protected sites, resulting in high biomass yields along with the considerable nutritional value of grasses and legumes.

**Keywords:** biomass production; chemical composition; ecological management; ecosystem diversity; grazing

**Citation:** Mugloo, J.A.; Khanday, M.u.d.; Dar, M.u.d.; Saleem, I.; Alharby, H.F.; Bamagoos, A.A.; Alghamdi, S.A.; Abdulmajeed, A.M.; Kumar, P.; Abou Fayssal, S. Biomass and Leaf Nutrition Contents of Selected Grass and Legume Species in High Altitude Rangelands of Kashmir Himalaya Valley (Jammu & Kashmir), India. *Plants* **2023**, *12*, 1448. https://doi.org/10.3390/ plants12071448

Academic Editors: Bingcheng Xu and Zhongming Wen

Received: 12 November 2022 Revised: 10 February 2023 Accepted: 20 March 2023 Published: 25 March 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

#### **1. Introduction**

The Jammu and Kashmir (J and K) region (India) is well known for its alpine and subalpine pastures [1], locally named "Margs" or "Bahaks". These pastures constitute a crucial ecological resource and play a significant role in the socioeconomic state of the Himalayas Valley. The total area of Kashmir pasturelands is around 9595 km2 [2]. These pasturelands are rural areas that involve around 97% of the population in the agricultural sector. The rearing of sheep, goats, and cattle constitutes the locals' subsidiary occupation. In addition, a huge population of nomadic Bakerwals, Gujjars, Chopans, Changpas, and Gaddies depends directly on meadow products and pasturelands for herds of maintained livestock.

The term "grassland" (also designated by "pastureland") can be defined as land (and the vegetation growing on it) devoted to the production of introduced or indigenous forage for harvest via grazing, cutting, or both. The grassland's vegetation includes grasses, legumes, and scantly woody species [3]. Thus, grassland is a highly dynamic ecosystem that supports fauna, flora, and human populations worldwide. It also encloses fodder crops that covered approximately 3.5 billion hectares in 2000 and contains around 20% of the world's soil carbon stocks [4]. These stocks can be well enriched through the good management of grasslands via Voisin's rational grazing (VRG), resulting in an increased milk production by ruminants [5]. This system allows the maximization of pasture growth associated with ruminant intake while maintaining a sustainable circular cycle [6]. The adoption of such a system is a must, especially in the Himalayan pasturelands, which was a scene of overgrazing over decades and centuries [7]. The literature reported a decrease in the available grazing area in the alpine and subalpine pasturelands of Kashmir from 0.15 ha. animal−1 to 0.10 ha. animal−1 between 1977 and 1982 [8]. Unfortunately, it is still at a continuous decrease rate according to a recent report [9].

As the globe is still facing a rise in climatic change, there is an increased demand to monitor, forecast and predict its effect on the productivity and quality of pasturelands [10]. However, the prediction of pasturelands' biomass is sometimes not very reliable, while the accurate and precise estimation consists of traditional methods that are mainly costly, time-consuming, and non-environmentally friendly. In addition to this, the traditional influencing factors (i.e., high wind velocity, low temperature, snowstorms, and so on) and the increase in solar and ultraviolet (UV) radiations—as a direct response to climate change—have resulted in the decreased biomass production of Himalaya pasturelands [11]. Similar observations were acknowledged in Nepal [12] and in the African pasturelands of Niger and Zambia [10,13]. Thus, since the crucial role of pastureland in ecosystem functioning and soil stability is endangered, the safeguarding of the Himalayan alpine and subalpine vegetation exerts its weight [14].

Although earlier studies focused on the aboveground herbaceous species production in a very limited area in Kashmir Valley [15], none have explored the whole region's above and belowground biomass (AGB and BGB, respectively), nor documented reliably the prominent grasses and legumes species growing there. These grasses and legumes are the main diet for sheep, goats, and cattle reared in the region. Thus, as the local population mainly relies on milk production, the compositional quality evaluation of biomass used for sheep, goats, and cattle nutrition is crucial. Therefore, the current study aimed to (a) investigate and estimate the biomass yield and leaf nutritional profile of grasses and legumes in high-altitude pasturelands of the northern, central and southern Kashmir Himalaya Valley (Jammu and Kashmir (J&K)), India; and (b) detect any possible effect (danger i.e., overgrazing) on grassland biomass in the studied zones.

#### **2. Results**

#### *2.1. Analysis of Above (AGB), Below Ground Biomass (BGB) and Total Biomass Yield in Northern Kashmir*

In northern Kashmir region, the average AGB/BGB ratio was the highest in the grazed and seed-sown sites of Kupwara district (1.61 and 1.85, respectively), and in the protected

sites of Baramulla district (1.38) (Table 1). The average ratio of protected sites biomass over grazed sites biomass (R1) was comparable between districts, whereas the average ratio of seed-sown sites' biomass over grazed sites' biomass (R2) was the highest in Kupwara district. The percentages of villages within districts, where AGB > BGB (P1), and seed-sown sites biomass > grazed sites biomass (P3), were the highest in Kupwara district (100%). All districts showed a comparable percentage (100%) of villages where protected sites biomass > grazed sites biomass (P2).

**Table 1.** Comparison between northern Kashmir districts in terms of biomass production.

AGB: above-ground biomass; BGB: below-ground biomass; R1: average ratio of protected sites biomass over grazed sites biomass (protected sites biomass/grazed sites biomass); R2: average ratio of seed-sown sites biomass over grazed sites biomass (seed-sown sites biomass/grazed sites biomass); P1: percentage of villages within district where ABG > BGB; P2: percentage of villages within district where protected sites biomass > grazed sites biomass; P3: percentage of villages within district where seed-sown sites biomass > grazed sites biomass. Values are means; means within the same row followed by different letters are significantly different at *p* < 0.05 according to Duncan's multiple range test.

All northern Kashmir districts showed higher AGB by 32.0%–43.2% in grazed sites compared to BGB, except in Firozpur (Baramulla district) and Ketsan (Bandipora district) where BGB was higher (*p* < 0.05) by 40.0% than AGB (Table 2). In protected sites, AGB was also higher (*p* < 0.05) by a range of 1.4%–40.3% than BGB, except in Ketsan (Bandipora district) where the latter was higher (*p* < 0.05) by 12.3% than the former. In all northern Kashmir districts, seed-grown sites showed higher (*p* < 0.05) AGB by a range of 20.1%– 56.4% than BGB, whereas BGB was more abundant than AGB (*p* < 0.05) by 15.3% and 12.1% in Firozpur (Baramulla district) and Ketsan (Bandipora district), respectively. It was also depicted that the average biomass was higher (*p* < 0.05) for AGB than BGB in all districts (24.4%–43.2%), except for Firozpur (Baramulla district) and Ketsan (Bandipora district) (14.8% and 18.6%, respectively). Rajwar (Kupwara district), Gulmarg (Baramulla district), and Cithernaar (Bandipora district) showed the highest AGB and BGB among all districts and studied sites of northern Kashmir region. On the other hand, Gulmarg (Baramulla district) had the highest (*p* < 0.05) AGB among all studied sites, while Rajwar (Kupwara district) and Cithernaar (Bandipora district) had the highest (*p* < 0.05) BGB.

**Table 2.** Above (AGB) and below Ground Biomass (BGB) (t/ha) in the pasture of northern Kashmir.

**Table 2.** *Cont.*

Values are means (3 replicates of each biomass type); means within the same column (comparison between villages in terms of AGB/BGB) followed by different capital letters are significantly different at *p* < 0.05 according to Duncan test; means within the same column (comparison between AGB and BGB within each village) followed by different lowercase letters are significantly different at *p* < 0.05 according to Student's t test.

#### *2.2. Analysis of Above (AGB), Below Ground Biomass (BGB) and Total Biomass Yield in Central Kashmir*

In central Kashmir region, the average AGB/BGB ratio was the highest in all studied sites of Budgam district (2.08, 1.72, and 2.11 at grazed, protected and seed-sown sites, respectively), (Table 3). R1 was comparable between Ganderbal and Srinagar districts (2.70), whereas R2 was the highest in Srinagar district (1.53). P1, P2, and P3 were comparable between all central Kashmir districts (100%).

AGB: above-ground biomass; BGB: below-ground biomass; R1: average ratio of protected sites biomass over grazed sites biomass (protected sites biomass/grazed sites biomass); R2: average ratio of seed-sown sites biomass over grazed sites biomass (seed-sown sites biomass/grazed sites biomass); P1: percentage of villages within district where ABG > BGB; P2: percentage of villages within district where protected sites biomass > grazed sites biomass; P3: percentage of villages within district where seed-sown sites biomass > grazed sites biomass. Values are means; means within the same row followed by different letters are significantly different at *p* < 0.05 according to Duncan's multiple range test.

All central Kashmir districts showed a higher AGB (*p* < 0.05) by 19.7%–70.2% in grazed sites than BGB (Table 4). In protected sites, AGB was also higher (*p* < 0.05) by a range of 12.8%–47.9% than BGB. In all central Kashmir districts, seed-grown sites showed higher (*p* < 0.05) AGB by a range of 18.7%–57.1% than BGB. Moreover, the average biomass was higher (*p* < 0.05) for AGB than BGB in all districts (20.8%–55.6%). Narang (Ganderbal district), Kanidajan (Budgam district), and Chirenbal (Srinagar district) showed the highest AGB and BGB among all districts and studied sites of central Kashmir region. On the other hand, Kanidajan (Budgam district) showed significantly higher (*p* < 0.05) AGB and BGB in all sites except protected ones.

**Table 4.** Above (AGB) and belowground biomass (BGB) (t/ha) in the pasture of central Kashmir.

Values are means (3 replicates of each biomass type); means within the same column (comparison between villages in terms of AGB/BGB) followed by different capital letters are significantly different at *p* < 0.05 according to Duncan test; means within the same column (comparison between AGB and BGB within each village) followed by different lowercase letters are significantly different at *p* < 0.05 according to the Student's t-test.

### *2.3. Analysis of Above (AGB), Below Ground Biomass (BGB) and Total Biomass Yield in Southern Kashmir*

In southern Kashmir region, the average AGB/BGB ratio was the highest in the grazed and seed-sown sites of Anantnag district (1.71 and 1.82, respectively), and the protected sites of Shopian district (1.38) (Table 5). R1 and R2, and P1 and P3 scored the highest values in Anantnag and Kulgam districts, respectively, (2.38 and 1.47, and 100%, respectively). P3 was comparable between all southern Kashmir districts (100%).

**Table 5.** Comparison between southern Kashmir districts in terms of biomass production.

AGB: above-ground biomass; BGB: below-ground biomass; R1: average ratio of protected sites biomass over grazed sites biomass (protected sites biomass/grazed sites biomass); R2: average ratio of seed-sown sites biomass over grazed sites biomass (seed-sown sites biomass/grazed sites biomass); P1: percentage of villages within district where ABG > BGB; P2: percentage of villages within district where protected sites biomass > grazed sites biomass; P3: percentage of villages within district where seed-sown sites biomass > grazed sites biomass. Values are means; means within the same row followed by different letters are significantly different at *p* < 0.05 according to Duncan's multiple range test.

Particularly, all southern Kashmir districts showed a higher AGB (*p* < 005) by 31.6%– 51.9% in grazed sites than BGB, except in Dabjan (Shopian district) and Astanmarg (Kulgam district) where BGB was higher (*p* < 0.05) by 30.6% and 42.0% than AGB, respectively

(Table 6). The same trend was observed regarding protected sites where AGB were higher (*p* < 0.05) by a range of 2.0%–40.3% than BGB, except in Dabjan (Shopian district) and Astanmarg (Kulgam district) where BGB was higher (*p* < 0.05) by 5.4% and 18.0% than AGB, respectively. In all southern Kashmir districts, seed-grown sites showed higher (*p* < 0.05) AGB by a range of 17.4%–56.6% than BGB, whereas BGB was more abundant (*p* < 0.05) by 15.3% and 12.8% than AGB in Dabjan (Shopian district) and Astanmarg (Kulgam district), respectively. Also, the average biomass was higher (*p* < 0.05) for AGB over BGB in all districts (14.9%–44.3%), except for Dabjan (Shopian district) and Astanmarg (Kulgam district) (15.1% and 22.0%, respectively). Aru Valley (Anantnag district), Kaller (Shopian district), and Chirenbal (Kulgam district) showed the highest AGB and BGB among all districts and studied sites. On the other hand, AGB was the highest (*p* < 0.05) in Kaller (Shopian district) in grazed and protected sites, and in Chirenbal (Kulgam district) among seed-sown sites. BGB was the highest (*p* < 0.05) in Chirenbal (Kulgam district) in grazed and seed-grown sites, and in Aru Valley (Anantnag district) in protected sites.

**Table 6.** Above (AGB) and belowground biomass (BGB) (t/ha) in the pasture of southern Kashmir.

Values are means (3 replicates of each biomass type); means within the same column (comparison between villages in terms of AGB/BGB) followed by different capital letters are significantly different at *p* < 0.05 according to Duncan test; means within the same column (comparison between AGB and BGB within each village) followed by different lowercase letters are significantly different at *p*< 0.05 according to the Student's t test.

### *2.4. Comparison of Above (AGB), Belowground Biomass (BGB) and Total Biomass Yield between Kashmir Regions*

The comparison between northern, central, and southern Kashmir regions revealed that AGB (86.74, 78.62, and 75.22 t. ha−1 ), BGB (52.04, 51.16, and 50.99 t. ha−1 ), and total biomass yield (138.78, 129.78, and 126.21 t. ha−1 ) were the highest in central Kashmir region, followed by southern and northern Kashmir ones, respectively, (Figure 1). More precisely, AGB and total biomass yield recorded the highest values in the protected sites of central Kashmir region (42.04 and 68.41 t. ha−1 , respectively), whereas BGB scored the highest value in the protected sites of southern Kashmir region (27.00 t. ha−1 ).

highest value in the protected sites of southern Kashmir region (27.00 t. ha−1).

The comparison between northern, central, and southern Kashmir regions revealed that AGB (86.74, 78.62, and 75.22 t. ha−1), BGB (52.04, 51.16, and 50.99 t. ha−1), and total biomass yield (138.78, 129.78, and 126.21 t. ha−1) were the highest in central Kashmir re‐ gion, followed by southern and northern Kashmir ones, respectively, (Figure 1). More pre‐ cisely, AGB and total biomass yield recorded the highest values in the protected sites of central Kashmir region (42.04 and 68.41 t. ha−1, respectively), whereas BGB scored the

**Figure 1.** AGB, BGB and total biomass yield (t. ha−1) in studied circles. **Figure 1.** AGB, BGB and total biomass yield (t. ha−1 ) in studied circles.

#### *2.5. Yield and Nutrient Profile of Prominent Grasses and Legumes 2.5. Yield and Nutrient Profile of Prominent Grasses and Legumes*

The results of prominent grasses' yield and nutrient profile evaluation are shown in Table 7. Orchard grass (*Dactylis glomerata*) showed the highest yield (*p* < 0.05) (12.50 ± 0.6 t. ha−1) compared to other grasses, whereas, the highest nitrogen content (*p* < 0.05) was detected in Timothy grass (1.81 ± 0.05%). Phosphorus and crude protein contents were the most abundant (*p* < 0.05) in Agrostis grass (*Agrostis alba*) (0.34 ± 0.04% and 10.40 ± 0.5%, respectively), whereas perennial ryegrass (*Lolium perenne*) enclosed the highest (*p* < 0.05) potassium and crude fiber contents (0.44 ± 0.05% and 35.12 ± 0.8%, respectively). It should be noted that no significant difference (*p* > 0.05) and very low standard deviations (SDs) (0.2 < SD < 0.6, and 0.01 < SD < 0.8) were observed in terms of grass yield, and leaf nutrient content, respectively, between all studied sites in the three Kashmir regions. The results of prominent grasses' yield and nutrient profile evaluation are shown in Table 7. Orchard grass (*Dactylis glomerata*) showed the highest yield (*p* < 0.05) (12.50 ± 0.6 t. ha−1 ) compared to other grasses, whereas, the highest nitrogen content (*p* < 0.05) was detected in Timothy grass (1.81 ± 0.05%). Phosphorus and crude protein contents were the most abundant (*p* < 0.05) in Agrostis grass (*Agrostis alba*) (0.34 ± 0.04% and 10.40 ± 0.5%, respectively), whereas perennial ryegrass (*Lolium perenne*) enclosed the highest (*p* < 0.05) potassium and crude fiber contents (0.44 ± 0.05% and 35.12 ± 0.8%, respectively). It should be noted that no significant difference (*p* > 0.05) and very low standard deviations (SDs) (0.2 < SD < 0.6, and 0.01 < SD < 0.8) were observed in terms of grass yield, and leaf nutrient content, respectively, between all studied sites in the three Kashmir regions.

1.81

1.23 ± 0.03c

± 0.05a

±0.03c

**Table 7.** Yield and leaf nutrient profile of prominent grasses. **Table 7.** Yield and leaf nutrient profile of prominent grasses.

8.57

± 0.4c

3.51 ± 0.2d

±0.2d

*Phleum pratense*

(Timothy grass)

*Agrostis alba* (Agrostis grass)

(Agrostis grass)

(Timothy grass) ±0.4c ±0.05a ±0.03c ±0.04b ±0.8a ±0.4a *Agrostis alba* 3.51 1.23 0.34 ±0.04a 0.23 34.20 10.40 Values are means (3 replicates of each grass); means within the same column followed by different letters are significantly different at *p*< 0.05 according to Duncan's multiple range test.

0.21

± 0.03c

0.34 ± 0.04a 0.39

± 0.04b

0.23 ± 0.03d

±0.03d

31.20

± 0.8a

34.20 ± 0.8a

±0.8a

9.50

10.40 ± 0.5a

± 0.4a

±0.5a

Table 8 showed that red clover (*Trifolium pretense*) had the highest (*p* < 0.05) yield, nitrogen, potassium, and crude protein contents (25.40 ± 0.6 t. ha−1 , 1.69 ± 0.05%, 0.45 ± 0.05%, and 22.70 ± 0.6%, respectively), among prominent legumes found in Kashmir Valley, whereas the phosphorus content was the most abundantly found (*p* < 0.05) in white clover (*Trifolium repens*) and sainfoin (*Onobrychis viciifolia*) (0.34 ± 0.04%). In addition, the crude fiber content scored its highest value in white clover (33.12 ± 0.8%) compared to other prominent legumes studied. It should be noted that no significant difference (*p* > 0.05) and very low SDs (0.3 < SD < 0.6, and 0.02 < SD < 0.8) were observed in terms of legume

yield and leaf nutrient content, respectively, between all studied sites in the three Kashmir regions.

**Table 8.** Yield and leaf nutrient profile of prominent legumes.

Values are means (3 replicates of each grass); means within the same column followed by different letters are significantly different at *p*< 0.05 according to Duncan's multiple range test.

#### **3. Discussion**

#### *3.1. Analysis of Above (AGB), Belowground Biomass (BGB), and Total Biomass Yield*

The healthy functioning of an ecosystem can be estimated by the overall plant biomass it yields. Over-grazing is one of the most critical issues facing plant biomass in Kashmiri grasslands. The current study detected higher AGB compared to BGB and a positive AGB/BGB ratio in all districts of northern, central, and southern Kashmir regions. This simulates that the studied locations were not over-grazed. However, the high BGB in Aru Valley, Kaller and Chirenbal (northern Kashmir), Rajwar, Gulmarg, and Cithernaar (southern Kashmir), and Naranag, Kanidajan, and Chirenbal (central Kashmir) might reveal possible moderate grazing within these regions. Such regions might be abandoned after being heavily grazed in elder decades. In this regard, Dai et al. [16] reported that the moderate grazing promoted the root biomass of *Kobresia* meadow (BGB) in the northern Qinghai-Tibet pastures, which corroborates with our findings. Thus, the adaptive response of plants might occur in which they tend to increase their root development to survive [17]. Also, during the April-May period, snow melting occurs in the pasturelands of Kashmir Valley which naturally favors and promotes the growth and development of BGB. Furthermore, the protected areas (sites) by local authorities may have helped in the preservation of plant biomass away from rearing and over-grazing, thus resulting in increased biomass yields. This is consistent with the report of Lone and Pandit [18] on the Langate Forest division of Kashmir. Tittonell et al. [19] proposed an agroecological research agenda, suggesting species breeding to preserve diversity and a co-innovation of large-scale farming with farmers, policymakers, and value chains. Although site protection resulted in substantial improvements in rangeland grazing management in Namibia, it did not enhance cattle productivity nor rangeland health [20]. The interesting observation in the present study is that all studied species were found in grazed, protected, and seed-sown sites, which outlines again that grasslands were not over-grazed in the studied locations. Two decades back, it was reported that dry matter biomass yield ranged between 1.41 and 6.23 t. ha−1 in temperate pastures of the northwestern Himalayas [21], which is far below that observed in the current study. This could be explained by the fact that Kashmiri citizens are now more conscious and aware of the risks on grassland biomass and the disequilibrium of the environmental balance associated with overgrazing. On the other hand, climatic contrasts, species heterogeneity, and anthropogenic disturbances were reported to affect the biomass yield in the lesser Himalayan foothills, and northwestern regions of Kashmir Valley [22,23].

This simulates a possible inclusion of non-native species to the studied sites, resulting in the variation of soil organic carbon (SOC), and thus a variation in biomass yields [24]. Our findings outlined that AGB, BGB, and total biomass yield were the highest in central Kashmir, followed by southern and northern Kashmir. Such a variation in biomass yields was reported to be correlated with the variation in carbon sequestration potential (CSP) [25]. Panwar et al. [25] reported a high CSP in northern India (J & K as a whole state), associated with high biomass yields (AGB: 6.7–159.4 t. ha−1 ; BGB: 1.6–71.5 t. ha−1 ; total biomass yield: 15.9–202.6 t. ha−1 ). However, the study was very general and did not take into consideration the difference between Kashmiri regions in terms of vegetative populations nor site types (grazed, protected and seed-sown sites). Despite that, the AGB, BGB, and total biomass yields observed in the present study fall within the ranges stated in the aforementioned ones.

#### *3.2. Yield and Nutrient Profile of Prominent Grasses and Legumes*

The nutritional composition of grasses used for animal forage is a factor determining their growth, reproduction, and livestock production. Additionally, the climate, soil type, and degree of maturity of grasses influence their nutritional composition [26,27]. Sampling was performed during July which means that the studied grasses and legumes were at their harvest stage [28]. This simulates that their nutrient richness may have started to decline [29]. On the other hand, Hao and He [30] outlined an increased biomass yield once a nutrient loss occurs. This statement partly agrees with our findings as biomass yield was satisfying while grasses were highly nutritious. Leaf N, P, and K contents in orchard grass was several folds higher than outlined in other grasslands (N: 0.22–0.26%, *p*: 0.002–0.004%, K: 0.007–0.02%) [31]. A previous study on Chinese seed-sown pastures outlined leaf P and K contents in tall fescue grass higher by 1.7-fold and 4.9-fold than our findings [32]. Generally, leaf nitrogen content in grasses should not exceed 3.5% [33]; thus, our findings showed safe values. Leaf phosphorus content in grasses does not usually exceed 0.7–0.8% [34], which agrees with our findings, whereas leaf potassium content can range between 1.2 and 2.0% [35]. This simulates that the selected grasses may show some K deficiency. Moreover, Chang et al. [32] depicted a crude protein content in the range of 11–14%, being 1.5–1.9-fold higher than observed in the present study. On the other hand, the crude protein and crude fiber contents in selected legumes was promising. It was recommended that 12–19% of crude protein would be suitable for cattle feed [36]; which means that some of the selected grasses can be mixed with red or white clover to improve the protein requirements for rearing cattle. Thus, the high management of grasslands is proposed to increase the crude protein and crude fiber contents in selected grasses. In this context, Berauer et al. [37] reported that crude protein in grasses increased by 22–30% after high land management. Furthermore, the moderate nitrogen percentages in the studied grasses revealed the absence of an over-grazing activity in the studied sites. Dong et al. [38] explained that increased N rates, N mineralization, and nitrification processes occur when associated with over-grazing. It is worth noting that animals have different behaviors based on their preference. For instance, cattle and sheep diets are mainly based on grasses and legumes while goats' diet is more related to herbal biomass [39]. Therefore, the present grasslands studied enclose nutritious grasses and legumes highly abundant for cattle and sheep foraging. This cannot be achieved without the inhibition of biomass species' eradication unless correct and serious management of lands is performed associated with new technology that is timesaving and has a lower negative impact on the environment.

#### **4. Materials and Methods**

#### *4.1. Study Area and Sites Description*

The study area enclosed the whole Jammu and Kashmir (J and K) Valley, which was divided into three zone circles: northern, central, and southern. Within each circle, three districts were selected, and subsequently, three sites within each district were chosen (Table 9). All studied sites varied in elevation between 1450 and 4800 m above mean sea

level. Studied areas enclosed: (a) Grazed Sites, (b) Protected Sites, and (c) Seed-Sown Sites. Grazed sites are grassland areas covered by grasses and legumes that are suitable for livestock grazing. Protected sites are grassland areas where carbon emissions from land use change are limited, and nutrient sapping is avoided. This included the cultivation of trees as windbreaks to reduce soil erosion and crop rotation to keep good nutrient availability in soil. Seed-sown sites are grassland areas which were seeded with native engendered species due to overgrazing or extensive agricultural exploitation. Seeded species included: orchard, tall fescue, perennial rye, and Agrostis grasses, and legumes such as: red clover, white clover, lucerne, sainfoin, and crown vetch. These species were sown in a randomized complete block design (RCBD) between mid-August and mid-September at a rate of 36 g seeds/m2 . A harrowing process was also practiced for initial soil preparation in order to optimize seedling establishment.

**Table 9.** Spatial distribution of studied sites.

#### *4.2. Climate*

The union territory of J and K, India (33◦170–37◦200 N latitude, 73◦250–80◦300 E longitude) comprises two main physical regions: Outer Himalayas with sub-tropical and intermediate climate (Jammu), and Inner Himalayas with a temperate climate (Kashmir). The climate varies considerably with altitude; it is mild and salubrious in lower altitudes but very cold in higher-ups. Spring is cool and rather wet. Regarding the Outer Himalayas (Jammu), average minimum and maximum temperatures vary between −11 ◦C and 33 ◦C during winter and summer, respectively. Autumn is bright and pleasant, while winter is extremely cold and experiences heavy snowfalls. Frost is experienced from the middle of November onwards. The main form of precipitation is snow in winter and some stray rains, and showering in spring. The Jammu region receives an average annual precipitation of about 1103 mm in the form of rain and snow for about 70 days. Unlike the Outer Himalayas, there is no distinct rainy season in the Inner Himalayas. The minimum temperature of the Kashmir region falls within −7 ◦C in winter and the maximum goes up to 35 ◦C in summer. It is characterized by a mean minimum temperature below 8 ◦C for more than six consecutive months per year. Mean annual minimum and maximum temperatures range between 6.68 and 19.31 ◦C, respectively, and mean annual soil temperature ranges between 8 and 15 ◦C, thus the area belongs to the mesic temperature regime. The mean annual rainfall in the Inner Himalayas is 710 mm and the soil in the studied area does not remain generally dry for more than 90 cumulative days. Hence, it belongs to the udic moisture regime.

### *4.3. Vegetation Diversity*

Several types of grasses and legumes growing in the studied Kashmir regions are outlined in Table 10.

**Table 10.** Diversity of grasses and legumes in the studied Kashmir regions.

#### *4.4. Sampling*

Samples were collected following a direct field plot harvest method [40]. Briefly, three transects, divided into two blocks (100 m distant from each other) were performed. Moreover, three quadrants of 1 m2 each were performed within each block. All grasses and legumes, within these quadrants, were collected from their roots by digging 5 cm2 pits up to a depth of 30 cm. Then, they were packed in ice-cooled-bags, transported directly to the laboratory for identification, and stored at a cool temperature for further analyses.

#### *4.5. Compositional Analyses*

Before analyses, the vegetative components including roots were washed thoroughly under a jet of running tap water to remove the attached soil. Then, they were dipped in diluted HCl (1 mL concentrated HCL.L−1 water) [41]. Further washing was performed with de-ionized water. Sampled species were first identified as grasses or legumes, then the above- and belowground biomass yields were estimated in t. ha−1 [41]. Afterward, the roots were removed, and the clean leaf samples were dried in a hot air-circulating oven at 105 °C for 24 h until a constant weight is obtained. Then, they were girded for nutrient analysis [41].

The nitrogen content was estimated using the micro Kjeldahl method [42]. The phosphorus content was determined following the vanado-molybdo-phosphoric Acid yellow colorimetric method [43]. Briefly, the dissolved reactive phosphorus reacted with ammonium molybdate under acid conditions. Hence, the molybdo-phosphoric acid was formed, and a yellow vanado-molybdo-phosphoric acid was obtained in the presence of vanadium. This corresponds to the phosphorus concentration. Such concentration was detected at a wavelength of 470 nm using Thermo Spectronic Helios Gamma UV (Hellma model:

178.712-QS, flow cell: 10 mm light path, inner optical volume: 30 L), connected to a Kipp & Zonen BD112 recorder [43].

The potassium content was determined using the FLAPHO flame photometer method [44]. Briefly, the instrument was warmed up for 10 min and distilled water was fed to the instrument. Then, the indicators were adjusted to 0 (reading). The concentrated standard solution was aspirated, and the readout was adjusted to 90 (on the uppermost scale). Afterward, the distilled water was aspirated, and the instrument read 0. All standards and standard solutions were aspirated, and results were recorded. Then, the calibration curves were drawn; the potassium concentration corresponded to the abscissa, whereas the instrument readouts corresponded to the ordinate. Finally, the potassium concentration was noted.

Using the acid–alkali digestion method, the residue after acid and alkaline digestion (determined gravimetrically) corresponded to the crude fiber content [45]. The crude protein content was calculated after the determination of the leaf nitrogen via the micro Kjeldahl method. The leaf nitrogen content value was multiplied by a coefficient factor of 6.25 [46]. All compositional elements were expressed as percentage (%) dry matter.


**6. Diseases and pest management**

Oilseed rape plants can suffer from a comprehensive list of enemies, starting with the viruses and finishing with mammalians. Their main fungal diseases are canker, light leaf spot, *alternaria*, and *sclerotinia* stem rot. The canker symptoms are leaf spotting, premature ripening and stem weakening in the autumn-winter period. The treatment of fungicides conazole or triazole is applied in late autumn and spring to fight this disease. The light leaf spot disease is caused by the fungus *Pyrenopeziza brassicae*. This disease can be recognised by speckles of white spore pustules on leaves, stems and pods, which become visible only after a period of dry weather. *Alternaria* fungi species causes leaf spots. It can penetrate the pods and infect the seed. *Sclerotinia* is dangerous when lower main stems are infected. Infection can stop the food and water supply to the canopy and cause the death of the plant. Broad-spectrum fungicides are used to control *alternaria* and *sclerotinia* development [1, 39, 40].

Also, oilseed rape can be attacked by a wide variety of insects.  CSFB is currently the main oilseed rape pest enemy. Adults of CSFB graze on young oilseed rape plants and can cause plant death. CSFB larvae also mine within the petioles and stems of plants. Lower chances of CSFB spread are considered in areas where oilseed rape has not been grown recently or nearby [3, 40].

Despite continuously developing CSFB management relying on a variable and complex set of alternative solutions, it has recently become more challenging to control. First, due to a reduction in effective chemical control options: resistance to pyrethroid sprays and the withdrawal of neonicotinoid seed treatments. Foliar application of pyrethroid insecticides was the most approachable chemical control of CSFB in oilseed rape. However, the resistance to pyrethroid insecticides is now widespread, and lessens the control's potential. Thus, when CSFB resistance to pyrethroid insecticides has developed, measures must be taken. Subsequent pyrethroid applications are recommended to avoid. This is to stop the selection for resistance and harming natural enemies [3, 41].

Otherwise, it is essential to follow the spray thresholds and cure a full field. The decrease in effective chemical options in oilseed rape has spurred the rise of CSFB. Reducing oilseed rape areas in certain regions is often associated with a higher prevalence of CSFB. Thus, crop protection requires innovation in this pest management. There is a need to find novel approaches to control CSFB, e.g., breeding varieties with greater resilience to CSFB, using biopesticides or others. Presently there are no varieties available to control any pest of oilseed rape. Little attention is paid to biopesticides to control CSFB have received. Biopesticides serve as a pest control option. They cause a minimal environmental impact, are specific to the target pest and can be a tool for resistance management [10, 40].

Oilseed rape is particularly vulnerable to pest damage at the early stages of growth. And it is far more tolerant to attack after the cotyledons have unfolded. The need for treatment can be determined by the number of shot-holing symptoms, leaf area eaten by beetles (**Figure 3**) or larvae and the plant's growth rate. Along with the adult beetle harm, the autumn, winter and spring CSFB larval assessment in leaf petioles and stem is present. It needs to be highlighted that the larval damage may be more economically harmful. Autumn larvae are more damaging, as their consumption of plant material last longer, and winter and spring larval invasion is likely to be less significant. If the risk is high –treatment at the first sign of attack is recommended to

**Figure 3.** *CSFB holing symptoms in oilseed rape.*

#### *Oilseed Rape: Biology, Use, Current Cultivation Issues and Agronomic Management DOI: http://dx.doi.org/10.5772/intechopen.109180*

be considered at the initial stages of growth. For the latter growth stages, such fields should be set up with traps, and the number of pests should be monitored not to exceed [1, 4, 6].

Effective chemistry is considered the best CSFB management option, however, no individual chemical or non-chemical approach is absolutely warranted. Thus, a combination of techniques is recommended for CSFB population suppression. Several alternative methods need to be mentioned. It is important not to damage natural enemy populations, e.g., ground beetles and parasitic wasps, by pesticide applications (especially broad-spectrum pyrethroids) and intensive cultivation. Great benefits were obtained in trials with trap crops. This approach uses trap crops to attract CSFB and simultaneously divert it away from oilseed rape. Afterwards, CSFB eggs or larvae die when the trap crop is destroyed. The disadvantages of this approach are that the use of relatively large areas of trap crops (at least 2 ha) is recommended and that the benefits can be variable. However, the trial approaches showed up to 88% reduced adult CSFB infestation and 76% reduced oilseed rape damage. It is recommended for a trap crop to be at an early growth stage at the end of August. This could make it more attractive to CSFB [40–42].

As an alternative solution, the selection of cultivars is recommended. The susceptibility or attractiveness of different varieties to CSFB has not been evidenced. Nevertheless, it is recommended to choose hardy varieties in the context of CSFB control. Such varieties can reach the four-leaf stage faster. This property makes the plant more tolerant to CSFB adult and larvae feeding damage [3, 4].

The maintenance of optimal soil humidity during crop germination is essential. Aiming to provide sufficient moisture levels, sowing dates should be adjusted regarding the soil humidity level and weather conditions and forecast. In addition to maintaining soil humidity and enriching the soil with organic matter, the cereal stubble and straw also have several benefits of their presence in the field for CSFB management. It is suggested that stubble and straw make it difficult for CSFB to locate the germinating oilseed rape and are useful as a support for spider web [1, 5].

The date of sowing is also shown to play a vital role in the CSFB management. Avoiding the alignment between beetle migration peak and the most susceptible crop growth stages is recommended. The sowing period of the highest risk from adult CSFB lasts from the end of August to early September. The early sowing ensures successful oilseed rape germination and establishment before the CSFB migration. The late sowing reduces an adult CSFB beetle feeding damage by moving the germination after the migration peak and reducing the larval invasion by slowed egg laying and development under cooler conditions. Studies have shown that a 10-fold reduction in larval invasion is obtained by a three-week postponement of sowing.

The optimisation of seedling density can also be used to reduce the risk of CSFB damage. To achieve final optimal plant density under the threat of CSFB invasion, the seed rate needs to be increased. And on the other hand, by decreasing the seed rate, plants may grow larger and more tolerant of larval feeding in the spring [6, 34].

Companion crop species are known to prevent or lessen crop damage by attracting the pest, improving soil, masking the crop from pests, or offering shelter for natural enemies. For example, mustards act as sacrificial plants that are eaten first. The research on CSFB management has detected significantly lower damage in two-leaf and five-leaf stage oilseed rape cultivated with a list of companion crop species. Buckwheat, legumes and *Brassicaceae* family members are known to play a preventing role in CSFB management. The companion crop should be sown about a week before oilseed rape and can be removed by frost or by the herbicide. It should also not outcompete oilseed rape [3, 7, 41].

As CSFB larvae generally settle in leaf petioles, trials have shown that managed defoliation reduces larval invasion by up to 55%. It has also been found that sheep grazing and topping/flailing are effective in reducing larval populations. The most beneficial is late defoliation before stem extension (November and December). Later defoliation could be dangerous due to a lack of time for plants to recover. It is also suggested that CSFB damage could also be reduced by the addition of organic amendments during the establishment period. The use of organic amendments may be helpful to for CSFB invasion management by improving plant growth, masking the crop, and repelling the pest. Moreover, crop tolerance to larval damage can also be enhanced by proper nutrition and plant growth regulation. It should be noted that the management tools proposed to control the CSFB spread are also useful in eliminating many other issues in oilseed rape cultivation [12, 40].


3.1. The formal series method of computing degenerate singular point quantities on center manifold

Let us consider the real n-dimensional systems with two zero eigenvalues and zero linear part as follows

#### Mutiple Hopf Bifurcation on Center Manifold http://dx.doi.org/10.5772/65674 11

$$\begin{cases} \frac{d\mathbf{x}}{dt} = (\delta \mathbf{x} - \mathbf{y})(\mathbf{x}^2 + \mathbf{y}^2)^q + \sum\_{k+j+1=2q+2}^{\infty} A\_{kj1} \mathbf{x}^k \mathbf{y}^1 \mathbf{u}^1 = X(\mathbf{x}, \mathbf{y}, \mathbf{u}),\\ \frac{d\mathbf{y}}{dt} = (\mathbf{x} - \delta \mathbf{y})(\mathbf{x}^2 + \mathbf{y}^2)^q + \sum\_{k+j+1=2q+2}^{\infty} B\_{kj1} \mathbf{x}^k \mathbf{y}^1 \mathbf{u}^1 = Y(\mathbf{x}, \mathbf{y}, \mathbf{u}),\\ \frac{d\mathbf{u}\_i}{dt} = -d\_i \mathbf{u}\_i + \sum\_{k+j+1=2}^{\infty} d\_{kj} \mathbf{z}^k w^j \mathbf{u}^1 = \mathcal{U}\_i(\mathbf{x}, \mathbf{y}, \mathbf{u}), \ i = 1, 2, \dots, n-2 \end{cases} \tag{23}$$

where the subscript "kj1" denotes "kjl1⋯ln−2", u1 ¼ ul1 1 ul2 2 ⋯uln−2 n−2, and l ¼ ∑ n−2 i¼1 li, all di > 0, x; y; ui; t; δ; Akjl; Bkjl; dkjl∈R, q; k; j; li∈N. Obviously, the origin of system (23) is a high-order degenerate singular point with two zero eigenvalues and n−2 negative ones.

In order to discuss the calculation method of the focal values on center manifold of the system (23), from the center manifold theorem [1], we take an approximation to the center manifold:

$$\mathbf{u} = \mathbf{u}(x, y) = \mathbf{u}\_2(x, y) + \mathbf{h}.\\ \mathbf{o}.\\ \text{t.} \tag{24}$$

where u ¼ ðx1;x2;⋯;xn−2Þ T, u2 is a quadratic homogeneous polynomial vector in x and y, and h. o.t. denotes the terms with orders greater than or equal to 3. Substituting u ¼ uðx;yÞ into the equations of system (23), we obtain a real planar polynomial differential system as follows

$$\begin{cases} \frac{\text{dx}}{\text{dt}} = (\delta \text{x} - y)(\text{x}^2 + y^2)^q + \sum\_{k=2q+2}^{\infty} \text{X}\_k(\text{x}, y) = \tilde{\text{X}}(\text{x}, y),\\ \frac{\text{dy}}{\text{dt}} = (\text{x} - \delta y)(\text{x}^2 + y^2)^q + \sum\_{k=2q+2}^{\infty} \text{Y}\_k(\text{x}, y) = \tilde{\text{Y}}(\text{x}, y) \end{cases} \tag{25}$$

where Xkðx;yÞ, Ykðx;yÞ are homogeneous polynomials of degree k, and the origin is degenerate with a zero linear part.

For system (25), some significant works have been done in Refs. [26] and [27]. Let us recall the related notions and results.

By means of transformation (5)

f 1 ¼ 8a3ce þ 8a3c−8a3e−8a3−2a2bce þ 2a2be þ 8a2ce þ 8a2c

ð2ae þ 2a−bÞðe þ 1),

d1 ¼ 8a3c−8a3−2a2bc þ 2a2b þ 8a2c−8a2 þ 3ab2 þ 3b2

f 4 ¼ 8a5c2−16a5c þ 8a5−2a4bc2 þ 2a4bc þ 8a4c2−16a4c þ 8a4 þ 2a3b2

−b3 ;

Theorem 6. For the flow on center manifold of the system (14), the first 2 focal values of the origin are

Remark 3. In contrast to the result and process in [36], one can easily see that our first quantity is basically consistent with its characteristic exponent of bifurcating periodic solutions, and our algorithm is easy to realize with computer algebra system due to the linear recursion formulas,

Theorem 7. At least two small limit cycles can be bifurcated from the origin of the 4D-hyoerchaotic

The rigorous proof of the above theorem is very similar to the previous ones in [14, 16], namely, by calculating the Jacobian determinant with respect to the functions v3; v5 and its variables,

Up till now, study on bifurcation of limit cycles from the degenerate singularity of higher dimensional nonlinear systems (1) is hardly seen in published references. Here, we will investigate the Hopf bifurcation problem from the high-order critical point on the center manifold.

3.1. The formal series method of computing degenerate singular point quantities on center

Let us consider the real n-dimensional systems with two zero eigenvalues and zero linear part

and more convenient to investigate the multiple Hopf bifurcation on center manifold.

Considering its Hopf bifurcation form of Theorem 6, we have the following:

system (14), which lie in the neighborhood of the origin restricted to the center manifold.

e þ 2ab2 þ 2abc−2ab þ 3b2

c þ 4a2b3 þ 2a2b2

Þðc−1Þ 3=2 ; c

;

v3 ¼ iπμ1; v5 ¼ iπμ2 (22)

e þ 3b2 ;

c

c þ 3ab2

−8a2e−8a2 þ ab2

d3 ¼ 9a2c−8a2 þ 2a þ 1;

3

d0 ¼ ða2c þ 2a þ 1Þð4a2c−4a2 þ b2

From Remark 1 and the singular point quantities (21), we have

where the expression of v5 is obtained under the condition of v3 ¼ 0.

f 3 ¼ 4a2e þ 4a2−3abe−2ab þ 4ae þ 4a þ b;

−4a3bc þ 4a3b−5a2b3

−2a2bc þ 2a2b−2ab3

d2 ¼ 8a2e þ 8a2−2abe þ 8ae þ 8a þ b2 þ 2b;

and the above expression of μ2 is obtained under the condition of μ1 ¼ 0.

f 2 ¼ ð2a þ b þ 2Þ

10 Manifolds - Current Research Areas

−2a3b2

−2a2b2

as follow

which will not be given here.

manifold

as follows

3. Case of the degenerate singular point

$$z = x + y\mathbf{i},\ \ w = x - y\mathbf{i},\ \ u = \nu,\ \ T = \mathbf{i}\ t,\ \ \mathbf{i} = \sqrt{-1},$$

system (25) is transformed into following system:

$$\begin{cases} \frac{d\mathbf{z}}{dT} = (\mathbf{1} \cdot \mathbf{i}\delta)\mathbf{z}^{q+1}w^{q} + \sum\_{k+j=2q+2}^{\infty} a\_{kj}\mathbf{z}^{k}w^{j} = \mathbf{Z}(\mathbf{z}, \mathbf{w}),\\ \frac{d\mathbf{w}}{dT} = -(\mathbf{1} + \mathbf{i}\delta)\mathbf{z}^{q}w^{q+1} - \sum\_{k+j=2q+2}^{\infty} b\_{kj}\mathbf{z}^{k}w^{j} = -\mathbf{W}(\mathbf{z}, \mathbf{w}) \end{cases} \tag{26}$$

where z; w; T are complex variables and for any positive integer k; j, we have akj ¼ bkj, then systems (25) and (26) are called concomitant.

For any positive integer k, we denote

$$f\_k(z, w) = \sum\_{\alpha + \beta = k} c\_{\alpha\beta} z^{\alpha} w^{\beta}$$

a homogeneous polynomial of degree k with c00 ¼ 1; ckk ¼ 0; k ¼ 1, 2,⋯.

Theorem 8 ([26, 27]). For system (26) with δ ¼ 0, we can derive successively the terms of the following formal series:

$$F(z, w) = zw \left[ 1 + \sum\_{m=1}^{\infty} \frac{f\_{m(2\eta+3)}(z, w)}{(z\varpi)^{m(\eta+1)}} \right] \tag{27}$$

such that

$$\frac{\partial F}{\partial T} = \frac{\partial F}{\partial z} Z - \frac{\partial F}{\partial w} W = (zw)^q \sum\_{m=1}^{\infty} \mu\_m (zw)^{m+1}.\tag{28}$$

Definition 3. If δ ¼ 0 holds, μm in expression (28) is called the mth singular point quantity at the degenerate singular point for system (26) or (1.3.26) is also called the mth singular point quantity of the origin on the center manifold of system (23), where m ¼ 1; 2;⋯:

Similar to Theorem 2, there also exists a equivalence between the mth singular point quantity and the mth focal value v2mþ1ð2πÞ at the origin on center manifold of system (23).

Theorem 9. For system (23) with δ ¼ 0, and any positive integer m, the following assertion holds: v2mþ1ð2πÞ~iπμm, namely

$$w\_{2m+1}(2\pi) = i\pi \left(\mu\_m + \sum\_{k=1}^{m-1} \xi\_m^{(k)} \mu\_k \right),\tag{29}$$

where ξðkÞ m ðk ¼ 1; 2;⋯;m−1Þ are polynomial functions of coefficients of system (26). Then, the relation between v2mþ1ð2πÞ and μm is called the algebraic equivalence.

Remark 4. In fact, from Theorem 2, for any positive integer m ¼ 2; 3;⋯, if μ1 ¼ μ2 ¼ ⋯ ¼ μm−1 ¼ 0 and v1ð2πÞ ¼ v3ð2πÞ⋯ ¼ v2m−1ð2πÞ ¼ 0 hold, and vice versa. And more the stability and bifurcation of the origin of system (23) can be figured out by calculating the singular point quantities.

Corollary 2. The origin of system (23) is a center restricted to the center manifold if and only if μm ¼ 0 for all m.

#### 3.2. An example of three-dimensional system

Now we consider an example for system (23) with n ¼ 3, it can be put in its concomitant form as follows

#### Mutiple Hopf Bifurcation on Center Manifold http://dx.doi.org/10.5772/65674 13

$$\begin{cases} \frac{d\mathbf{z}}{dT} = (\mathbf{1} \cdot \mathbf{i}\delta)\mathbf{z}^2 w + \imath\omega \left(a\_{20}\mathbf{z}^2 + a\_{11}\varpi w + a\_{02}w^2\right) = \mathbf{Z},\\ \frac{d\mathbf{w}}{dT} = -(\mathbf{1} + \mathbf{i}\delta)\varpi w^2 - \imath\omega w \left(b\_{20}w^2 + b\_{11}\imath\varpi + b\_{02}\mathbf{z}^2\right) = -\mathcal{W},\\ \frac{d\mathbf{u}}{dT} = \mathbf{i}\mathbf{u} + \mathbf{i}d\_1\varpi w = \mathcal{U}, \end{cases} \tag{30}$$

where d1≠0 and

f kðz;wÞ ¼ ∑

a homogeneous polynomial of degree k with c00 ¼ 1; ckk ¼ 0; k ¼ 1, 2,⋯.

dF dT ¼ ∂F ∂z

between v2mþ1ð2πÞ and μm is called the algebraic equivalence.

3.2. An example of three-dimensional system

Fðz;wÞ ¼ zw 1 þ ∑

Z− ∂F ∂w

quantity of the origin on the center manifold of system (23), where m ¼ 1; 2;⋯:

and the mth focal value v2mþ1ð2πÞ at the origin on center manifold of system (23).

v2mþ1ð2πÞ ¼ iπ μm þ ∑

following formal series:

12 Manifolds - Current Research Areas

v2mþ1ð2πÞ~iπμm, namely

where ξðkÞ

quantities.

for all m.

as follows

such that

αþβ¼k

Theorem 8 ([26, 27]). For system (26) with δ ¼ 0, we can derive successively the terms of the

∞ m¼1

W ¼ ðzwÞ

Definition 3. If δ ¼ 0 holds, μm in expression (28) is called the mth singular point quantity at the degenerate singular point for system (26) or (1.3.26) is also called the mth singular point

Similar to Theorem 2, there also exists a equivalence between the mth singular point quantity

Theorem 9. For system (23) with δ ¼ 0, and any positive integer m, the following assertion holds:

cαβzαwβ

f mð2qþ3Þðz;wÞ ðzwÞ

" #

q ∑ ∞ m¼1

> m−1 k¼1 ξðkÞ m μk

m ðk ¼ 1; 2;⋯;m−1Þ are polynomial functions of coefficients of system (26). Then, the relation

Remark 4. In fact, from Theorem 2, for any positive integer m ¼ 2; 3;⋯, if μ1 ¼ μ2 ¼ ⋯ ¼ μm−1 ¼ 0 and v1ð2πÞ ¼ v3ð2πÞ⋯ ¼ v2m−1ð2πÞ ¼ 0 hold, and vice versa. And more the stability and bifurcation of the origin of system (23) can be figured out by calculating the singular point

Corollary 2. The origin of system (23) is a center restricted to the center manifold if and only if μm ¼ 0

Now we consider an example for system (23) with n ¼ 3, it can be put in its concomitant form

� �

mðqþ1Þ

μmðzwÞ

mþ1

(27)

: (28)

, (29)

$$a\_{i\rangle} = A\_i + \mathbf{i}B\_i, \ b\_{i\rangle} = A\_i \mathbf{-i}B\_i, A\_i, B\_i \in \mathbb{R}, \, i, j = 0, 1, 2,\tag{31}$$

namely, aij ¼ bij. Then for the center manifold of system (30), from the transformation (5), we can determine the formal expression (24): u ¼ uðx;yÞ ¼ u~ðz;wÞ, thus obtain

$$\begin{cases} \frac{d\mathbf{z}}{dT} = (\mathbf{1} \cdot \mathbf{i} \delta) \mathbf{z}^2 w + \check{u} z \left( a\_{20} \mathbf{z}^2 + a\_{11} \mathbf{z} w + a\_{02} w^2 \right) = \check{\mathbf{Z}},\\ \frac{d\mathbf{w}}{dT} = -(\mathbf{1} + \mathbf{i} \delta) \mathbf{z} w^2 - \check{u} w \left( b\_{20} w^2 + b\_{11} w \mathbf{z} + b\_{02} \mathbf{z}^2 \right) = -\check{\mathbf{W}} \end{cases}.\tag{32}$$

Remark 5. For system (32), the corresponding n ¼ 1 in (27) and (28) of Theorem 8, we figure out that each μm is related to only the coefficients of the first 2m þ 3 order terms of system (32), m ¼ 1; 2;⋯. Here, we determine the above u~ just to the sixth-order term as follows

$$
\tilde{\boldsymbol{\mu}}(\boldsymbol{z}, \boldsymbol{w}) = \sum\_{k=2}^{6} \tilde{\boldsymbol{\mu}}\_{k}(\boldsymbol{z}, \boldsymbol{w}) \tag{33}
$$

where u~k is a homogeneous polynomial in z;w of degree k and

$$\begin{aligned} \|\boldsymbol{\tilde{u}}\_{2} = -d\_{1}zw, \ \boldsymbol{\tilde{u}}\_{4} = 2\delta d\_{1}z^{2}w^{2}, \ \boldsymbol{\tilde{u}}\_{3} = \boldsymbol{\tilde{u}}\_{4} = \boldsymbol{\tilde{u}}\_{5} = \boldsymbol{0}, \\ \boldsymbol{\tilde{u}}\_{6} = -\mathbf{i}d\_{1}zwz( (a\_{02} - b\_{20})d\_{1}w^{3}z + (a\_{11}d\_{1} - b\_{11}d\_{1} - 8\mathbf{i}\delta^{2})w^{2}z^{2} \\ + (a\_{20} - b\_{02})d\_{1}wz^{3}). \end{aligned} \tag{34}$$

Hence, Z~ and W~ in system (32) are two polynomials with degree 9.

Theorem 10. For system (32) with δ ¼ 0, we can derive successively the terms of the formal series (27), such that (28) holds (cαβ, μm in Appendix A).

Applying the powerful symbolic computation function of the Mathematica system and the recursive formulas in Theorem 10, and from Remark 5, we obtain the first three singular point quantities as follows

$$\begin{aligned} \mu\_1 &= -d\_1(a\_{11} - b\_{11}), \\ \mu\_2 &= d\_1^2(b\_{20}b\_{02} - a\_{20}a\_{02}), \\ \mu\_3 &= -2\text{id}\_1^2(a\_{02}a\_{20} + b\_{02}b\_{20} - a\_{02}b\_{02} - a\_{20}b\_{20}) \end{aligned} \tag{35}$$

In the above expression of each μk; k ¼ 2; 3, we have already let μ1 ¼ ⋯ ¼ μk−1 ¼ 0.

Thus, from Theorem 9 and Eqs. (35) and (31), we have

Theorem 11. For the flow on center manifold of system (30),δ¼0, the first three focal values v2iþ1ð2πÞ ði ¼ 1; 2; 3Þ of the origin are as follows

$$\begin{aligned} \upsilon\_3 &= 2\pi d\_1 B\_1, \\ \upsilon\_5 &= 2\pi d\_1^2 \left( A\_2 B\_0 + A\_0 B\_2 \right), \\ \upsilon\_7 &= 2\pi d\_1^2 \left[ \left( A\_0 - A\_2 \right)^2 + \left( B\_0 + B\_2 \right)^2 \right] \end{aligned} \tag{36}$$

Theorem 12. For the flow on center manifold of (30)δ¼0, the origin is a three-order weak focus, i.e., v3 ¼ v5 ¼ 0; v7 ≠ 0 if and only if

$$B\_1 = 0, A\_2 B\_0 + A\_0 B\_2 = 0 \text{ and } \left(A\_0 \text{--} A\_2\right)^2 + \left(B\_0 + B\_2\right)^2 \neq 0 \tag{37}$$

Remark 6. For the coefficients of system (30)δ¼0, there exists necessarily a group of critical values: Ai ¼ A� i ; Bi ¼ B� i ði ¼ 0; 1; 2Þ such that the conditions (37) hold, for example:

$$A\_1^\* = B\_1^\* = 0, A\_0^\* = B\_0^\* = 1, B\_2^\* = -A\_2^\* = 13\tag{38}$$

Now we consider Hopf bifurcation of limit cycles from the origin for perturbed system (30).

Theorem 13. At least three limit cycles can be bifurcated from the origin of system (30) restricted to the center manifold, which lie in the neighborhood of the origin.

Proof. From Theorem 11, one can easily calculate the Jacobian determinant with respect to the functions v3;v5;v7 and variables B1;B0;A0,

$$f = \frac{\partial(\upsilon\_3, \upsilon\_5, \upsilon\_7)}{\partial(B\_1, B\_0, A\_0)} = -2\pi^3 d\_1^5 [8(A\_0 A\_2 - A\_2^2 - B\_0 B\_2 - B\_2^2)]\tag{39}$$

Considering the conditions (37) of Theorem 12 and substituting the group of critical values of Eq. (38) into Eq. (39), we obtain J ¼ 649π3d5 1 ≠ 0. Thus, we take some appropriate perturbations for the coefficients of system (32) to make the following two conditions:

$$(\upsilon\_1(2\pi) - 1)\upsilon\_3 < 0, \upsilon\_3\upsilon\_5 < 0, \upsilon\_5\upsilon\_7 < 0\tag{40}$$

and

$$|e^{2\pi b} - 1| \ll |\upsilon\_3| \ll |\upsilon\_5| \ll |\upsilon\_7| \tag{41}$$

hold, one must obtain that the succession function on the center manifold has three small real positive roots, just the system (30) has at least three limit cycles in the neighborhood of the origin. We can refer to references [16, 26, 27] for more details about the construction of limit cycles.

Remark 7. In general, in order to find more limit cycles in the neighborhood of the origin of system (30), we should add more higher order terms of u~ðz;wÞ determined in Eq. (33). Here we propose a conjecture that system (30) has at most three limit cycles through Hopf bifurcation restricted to a center manifold from the origin. However, the center conditions or integrability at the degenerate singularity will need further study.


Acoustic Waves From Microdevices to Helioseismology

*Edited by Marco G. Beghi*

**ACOUSTIC WAVES**

**HELIOSEISMOLOGY**

Edited by **Marco G. Beghi**

**INTECHOPEN.COM**

**– FROM MICRODEVICES TO**


# **Synthesis and Characterization of Biomedical Materials**

Edited by

Leszek Adam Dobrzański, Anna D. Dobrzańska-Danikiewicz, Lech Bolesław Dobrzański and Joanna Dobrzańska

> Printed Edition of the Special Issue Published in *Processes*

www.mdpi.com/journal/processes


*3.1. Land-Use Conflict Measurement Methods*

The spatial conflict caused by human activities is in essence a game of spatial resource occupation by various land users. Spatial conflict is accompanied by the utilization of land resources, resulting in the change of regional spatial pattern and function, which further affects the original physical, chemical and ecological processes and affects the nature conditions [33,34].

Referring to previous studies [35,36], we established a measurement model for landuse conflict based on the conceptual model of ecological risk assessment [33,34,37]. In terms of describing the pressure from human disturbance and deterioration of natural conditions, the ability of the land resources themselves to withstand conflict pressure, and the stability of the land system, three indexes including the land complexity index, land fragility index and land stability index were selected to evaluate the land-use conflict [34,38,39]. Mathematical linear models were used to characterize a comprehensive index of land-use conflict (*CCI*):

$$\text{CCI} = \text{LCI} + \text{LFI} - \text{LSI} \tag{1}$$

In this formula, *CCI* refers to the comprehensive index of land use conflict; *LCI* refers to the land complexity index; *LFI* refers to the land fragility index; and *LSI* refers to the land stability index. Since land use spatial conflict is a complex scientific problem, it is difficult to determine the impact of complexity, vulnerability and stability on land-use spatial conflict. Therefore, this study uses equal weight to calculate the comprehensive index of land use spatial conflict without considering its possible nonlinear influence.


*3.1. Visualization Characterization*

The visualization results of Model I are shown in Figure 5. The left panel shows that, after a 500 ps canonical system synthesis simulation, the nanoparticles gradually leaned towards the center through the attraction of their own branched chains and the repulsion of the branched chains by water. The right panel of Figure 5 shows that the nanoparticles are well-dispersed among themselves and show different characteristics from conventional surface-active multimolecular aggregation.

**Figure 5.** Schematic diagram of the initial model: (**a**) Dispersed morphology, (**b**) NGO distribution.

The visualization model of Model II is shown in Figure 6a: the simulation process of NGOs goes through two processes. The first process is the transport of water solution towards the oil–water interface. The second process is the spontaneous orientation of the lipophilic end towards the oil phase, and the spontaneous orientation of the hydrophilic end towards the water phase. The regular self-aggregation at the oil–water interface is shown in Figure 6b.

**Figure 6.** Self-aggregation equilibrium configuration of the system: (**a**) Self-aggregation process, (**b**) self-assembled membrane configuration.