**3. Isotherm models: theoretical knowledge**

Isotherm models are used to determine an adsorbent's maximal sorption capacity and is represented in terms of the amount of metal absorbed per unit mass of adsorbent.

#### **3.1 One parameter model**

The simple adsorption isotherm model has only one parameter to explain the adsorption mechanism. The most basic adsorption isotherm is one in which the amount of solute adsorbed is proportional to the equilibrium effluent concentration [16]. Eq. (3) is the model.

$$\mathbf{q}\_{\rm eq} = \mathbf{K} \mathbf{C}\_{\rm eq} \tag{3}$$

#### **3.2 Two parameter models**

This section focuses on gaining theoretical insight into models with two parameters that explain the adsorption mechanism. Henry's law with constant, Langmuir, Freundlich, Dubinin-Radushkevich, Temkin, Hill-de Boer, Fowler-Guggenheim, Flory-Huggins, Halsey, Harkin-Jura, Jovanovic, Elovich, and Kiselev are among the 13 models covered.

### *3.2.1 Henry's law with intercept isotherm model*

This model was created to address the contradiction highlighted by the one parameter model and to be applicable over a wide range of metal ion concentrations [16]. Eq. (4) describes the model.

$$\mathbf{q}\_{\rm eq} = \mathbf{K} \mathbf{C}\_{\rm eq} + \mathbf{C} \tag{4}$$

#### *3.2.2 Langmuir isotherm model*

The Langmuir model assumes a homogenous surface and explains how the adsorbate forms monolayer coverage on the adsorbent's outer surface [17, 18]. The rate of adsorption is proportional to the percentage of open adsorbent surface, and the rate of *Removal of Divalent Nickel from Aqueous Solution Using Blue Green Marine Algae… DOI: http://dx.doi.org/10.5772/intechopen.103940*

desorption is related to the fraction of covered adsorbent surface. The model is described in Eq. (5).

$$\mathbf{q\_{eq}} = \frac{\mathbf{q\_{max}} \mathbf{b\_L C\_{eq}}}{\mathbf{1} + \mathbf{b\_L C\_{eq}}} \tag{5}$$

bL is the Langmuir constant, which links the fluctuation of the appropriate area and porosity of the adsorbent with its adsorption capacity (mg/g). A dimensionless constant called the Langmuir separation factor RL, which is computed as Eq. (6), helps explain the key properties of the Langmuir isotherm.

$$\mathbf{R}\_{\rm L} = \frac{1}{\mathbf{1} + \mathbf{b}\_{\rm L} \mathbf{q}\_{\rm max}} \tag{6}$$

When RL > 1, adsorption is unfavourable, when RL = 1, linear when RL = 1, favourable when RL = 0, and irreversible when RL = 0.

#### *3.2.3 Freundlich isotherm model*

The Freundlich adsorption isotherm model depicts the adsorbent surface heterogeneity. The adsorptive sites are made up of tiny homogenous heterogeneous adsorption sites [19]. Eq. (7) is the model.

$$\mathbf{q\_{eq}} = \mathbf{a\_F C\_{eq}^{\rm tr}} \tag{7}$$

Freundlich adsorption capacity (L/mg) is denoted by aF, while adsorption intensity is denoted by nF. The higher the adsorption capacity, the larger the aF value. The magnitude of 1/nF, which varies from 0 to 1, indicates favourable adsorption and becomes more heterogeneous as it approaches zero [18–21].

### *3.2.4 Dubinin-Radushkevich isotherm model*

This empirical model implies that physical adsorption processes are multilayered and involve Van Der Waal's forces [22], it is frequently used to estimate the characteristic porosity. This model [23] gives insight into gas and vapour adsorption on micro porous sorbents.

The Dubinin-Radushkevich isotherm's temperature dependence is another distinguishing trait [24, 25]. Eq. (8) represents the Dubinin-Radushkevich isotherm.

$$\mathbf{q\_{eq}} = \mathbf{K\_{DR}} \exp\left[-\mathbf{B\_{DR}} \left(\mathbf{RTIn} \left(\mathbf{1} + \frac{\mathbf{1}}{\mathbf{C\_{eq}}}\right)\right)^2\right] \tag{8}$$

$$\mathbf{e} = \text{RTln}\left[\frac{\mathbf{1}}{\mathbf{C\_{eq}}}\right] \tag{9}$$

ε is known as the Polanyi potential, as seen in Eq. (9). The activation energy or mean free energy E (kJ/mol) of adsorption per molecule of adsorbate when it is transported from infinity in the solution to the surface of the solid may be computed using Eq. (10).

$$\mathbf{E} = \frac{1}{\sqrt{2\mathbf{K}\_{\rm DR}}}\tag{10}$$

The value of E is used to forecast whether an adsorption is physical or chemical. In nature, physisorption occurs when E = 8 KJ/mol, whereas chemisorption occurs when E=8–16 KJ/mol [23].

#### *3.2.5 Temkin isotherm model*

For forecasting the gas phase adsorption equilibrium, the Temkin adsorption isotherm model is quite useful [25, 26]. Eq. (11) illustrates the Temkin adsorption isotherm model.

$$\mathbf{q}\_{\rm eq} = \frac{\mathbf{RT}}{\mathbf{b}\_{\rm T}} \left( \ln \mathbf{A}\_{\rm T} \mathbf{C}\_{\rm eq} \right) \tag{11}$$

The heat of adsorption is represented by the equation RT/bT, and the equilibrium binding constant (L/mg) corresponding to the maximal binding energy is represented by AT.

#### *3.2.6 Hill-de Boer isotherm model*

This Hill-Deboer isotherm model accurately describes mobile adsorptions as well as lateral interactions between adsorbed molecules [27, 28]. Eq. (12) depicts the Hill-Deboer isotherm model.

$$\mathbf{K}\_{\mathbf{1}} \mathbf{C}\_{\mathbf{eq}} = \frac{\boldsymbol{\theta}}{\mathbf{1} - \boldsymbol{\theta}} \cdot \exp\left(\frac{\boldsymbol{\theta}}{\mathbf{1} - \boldsymbol{\theta}} - \frac{\mathbf{K}\_2 \boldsymbol{\theta}}{\mathbf{RT}}\right) \tag{12}$$

A positive value of K2 implies attraction between adsorbed species, whereas a negative value of K2 suggests repulsion. If K2 is equal to zero, there is no interaction between adsorbed molecules, and the Volmer equation [29] is used

#### *3.2.7 Fowler-Guggenheim isotherm model*

The Fowler-Guggenheim adsorption isotherm model describes how adsorbed molecules interact laterally. Eq. (13) represents the Fowler-Guggenheim adsorption isotherm.

$$\mathbf{K\_{FG}C\_{eq}} = \frac{\boldsymbol{\Theta}}{\mathbf{1} - \boldsymbol{\Theta}} \exp\left[\frac{2\boldsymbol{\Theta}\mathbf{W}}{\mathbf{RT}}\right] \tag{13}$$

The presence of a positive W indicates that the contact between the adsorbed molecules is attractive. In contrast, if W = 0, the contact between adsorbed molecules is repulsive, and the heat of adsorption decreases with loading, the Fowler-Guggenheim equation reduces to the Langmuir model. W = 0 when there is no contact between adsorbed molecules. Furthermore, this model is only viable when the surface coverage is less than 0.6 [30, 31].

*Removal of Divalent Nickel from Aqueous Solution Using Blue Green Marine Algae… DOI: http://dx.doi.org/10.5772/intechopen.103940*

#### *3.2.8 Flory-Huggins isotherm model*

The degree of surface coverage of the adsorbate onto the adsorbent is discussed by the Flory-Huggins isotherm [32–35]. Eq. (14) shows the Flory-Huggins adsorption isotherm.

$$\frac{\Theta}{\mathbf{C}\_{\rm in}} = \mathbf{K}\_{\rm FH} \left[ \mathbf{1} - \boldsymbol{\theta} \right]^{\rm nF} \tag{14}$$

KFH is used to calculate the spontaneity Gibbs free energy.

#### *3.2.9 Halsey isotherm model*

In multi layer adsorption, the hetero porous nature of the adsorbent is demonstrated by the fitting of experimental data to this equation [36]. Eq. (15) denotes the Halsey adsorption isotherm model.

$$\mathbf{q}\_{\rm eq} = \exp\left[\frac{\ln|\mathbf{K}\_{\rm Ha} - \ln|\mathbf{C}\_{\rm eq}|}{\mathbf{n}\_{\rm Ha}}\right] \tag{15}$$

#### *3.2.10 Harkin-Jura isotherm model*

The Hurkin-Jura adsorption isotherm is used to explain multilayer adsorption on the surface of absorbents with heterogeneous pore distribution [37]. Eq. (16) describes the Hurkin-Jura adsorption isotherm model.

$$\mathbf{q\_{eq}} = \sqrt{\frac{\mathbf{A\_{Hj}}}{\mathbf{B\_{Hl}} - \log \mathbf{C\_{eq}}}} \tag{16}$$

#### *3.2.11 Jovanovic isotherm model*

With the approximation of monolayer localised adsorption without lateral contacts, the Jovanovic adsorption isotherm model is analogous to the Langmuir model [38]. Eq. (17) shows the Jovanovic adsorption isotherm model.

$$\mathbf{q}\_{\rm eq} = \mathbf{q}\_{\rm max} \left[ \mathbf{1} - \mathbf{e}^{-\left(\mathbf{k}\_{\rm l} \mathbf{C}\_{\rm eq}\right)} \right] \tag{17}$$

It forms the Langmuir isotherm at large adsorbate concentrations but does not obey Henry's rule.

#### *3.2.12 Elovich isotherm model*

Adsorption sites expand exponentially with adsorption, demonstrating multilayer adsorption, according to the Elovich isotherm model [39]. The Elovich adsorption isotherm model is depicted in Eq. (18).

$$\frac{\mathbf{q\_{eq}}}{\mathbf{q\_{max}}} = \mathbf{K\_{E}C\_{eq}} \exp^{\frac{q\_{eq}}{q\_{max}}} \tag{18}$$

#### *3.2.13 Kiselev isotherm model*

The Kiselev adsorption isotherm model is also known as localised monomolecular layer model [40]. This model is valid only for surface coverage θ > 0.68. The Kiselev adsorption isotherm model is given in Eq. (19).

$$\mathbf{K\_{eqK}} \mathbf{C\_{eq}} = \frac{\boldsymbol{\theta}}{\left(\mathbf{1} - \boldsymbol{\theta}\right)\left(\mathbf{1} + \mathbf{K\_{nK}}\boldsymbol{\theta}\right)}\tag{19}$$

#### **3.3 Three parameter models**

Models containing three parameters to explain the mechanism of adsorption are discussed using 16 models viz., Hill, Redlich-Peterson, Sips, Langmuir-Freundlich, Fritz-Schlunder-III, Radke-Prausnits, Toth, Khan, Koble-Corrigan, Jossens, Jovanovic-Freundlich, Brouers-Sotolongo, Vieth-Sladek, Unilan, Holl-Krich and Langmuir-Jovanovic.

#### *3.3.1 Hill isotherm model*

To characterise the adhesion of diverse species to homogeneous substrates, the Hill isotherm model is developed [41]. Eq. (20) depicts the Hill adsorption isotherm model.

$$\mathbf{q\_{eq}} = \frac{\mathbf{q\_{max}} \mathbf{C\_{eq}^{n\_H}}}{\mathbf{K\_H} + \mathbf{C\_{eq}^{n\_H}}} \tag{20}$$

If nH is greater than 1, this isotherm indicates positive co-operativity in binding, nH is equal to 1, it indicates non-cooperative or hyperbolic binding and nH is less than 1, indicating negative co-operativity in binding.

#### *3.3.2 Redlich-Peterson isotherm model*

The Redlich-Peterson isotherm model is created by combining elements of the Langmuir and Freundlich isotherms [42]. Eq. (21) depicts the Redlich-Peterson isotherm model.

$$\mathbf{q}\_{\rm eq} = \frac{\mathbf{A}\_{\rm RP} \mathbf{C}\_{\rm eq}}{\mathbf{1} + \mathbf{B}\_{\rm RP} \mathbf{C}\_{\rm eq}^{\beta}} \tag{21}$$

When the liquid phase concentration is low, this model approaches Henrys Law. βRP, the exponent, is usually between 0 and 1. When βRP = 1, this model is similar to the Langmuir model, and when βRP = 0, this isotherm is similar to the Freundlich model.

#### *3.3.3 Sips isotherm model*

The Sips adsorption isotherm model [43] was designed to represent localised adsorption without adsorbate-adsorbate interactions [44] at high adsorbate concentrations. Eq. (22) is the Sips adsorption isotherm model.

*Removal of Divalent Nickel from Aqueous Solution Using Blue Green Marine Algae… DOI: http://dx.doi.org/10.5772/intechopen.103940*

$$\mathbf{q}\_{\rm eq} = \frac{\mathbf{q}\_{\rm m} \mathbf{K}\_{\rm s} \mathbf{C}\_{\rm eq}^{\theta\_{\rm s}}}{\mathbf{1} + \mathbf{K}\_{\rm s} \mathbf{C}\_{\rm eq}^{\theta\_{\rm s}}} \tag{22}$$

When βS equal to 1 this isotherm approaches Langmuir isotherm and βS equal to 0, this isotherm approaches Freundlich isotherm.

#### *3.3.4 Langmuir-Freundlich isotherm model*

Adsorption in heterogeneous surfaces is described by the Langmuir-Freundlich isotherm model [17, 18]. Eq. (23) depicts the Langmuir-Freundlich isotherm model.

$$\mathbf{q}\_{\rm eq} = \frac{\mathbf{q}\_{\rm max} \left(\mathbf{K}\_{\rm LF} \mathbf{C}\_{\rm eq}\right)^{\rm m\_{\rm LF}}}{\mathbf{1} + \left(\mathbf{K}\_{\rm LF} \mathbf{C}\_{\rm eq}\right)^{\rm m\_{\rm LF}}} \tag{23}$$

mLF is a heterogeneous parameter with a value between 0 and 1. This value rises when the degree of surface heterogeneity decreases. For mLF is equal to 1, this model covert to Langmuir model.

#### *3.3.5 Fritz-Schlunder-III isotherm model*

Because of the large number of coefficients in their isotherm, the Fritz-Schunder three parameter isotherm model was constructed to suit a wide variety of experimental findings [45]. Eq. (24) has this expression.

$$\mathbf{q}\_{\rm eq} = \frac{\mathbf{q}\_{\rm max} \mathbf{K}\_{\rm FS3} \mathbf{C}\_{\rm eq}}{\mathbf{1} + \mathbf{q}\_{\rm max} \mathbf{C}\_{\rm eq}^{\rm m\_{FS3}}} \tag{24}$$

If mFS3 is equal to 1, the Fritz-Schlunder-III model becomes the Langmuir model but for high concentrations of adsorbate, the Fritz-Schlunder-III reduces to the Freundlich model.

#### *3.3.6 Radke-Prausnits isotherm model*

At low adsorbate concentrations, the Radke-Prausnits isotherm model has numerous essential qualities that make it preferable in most adsorption systems [46]. When the Radke-Prausnits model exponent mRaP3 is equal to zero. Eqs. (25)–(27) show Radke-Prausnits isotherm models.

$$\textbf{Model 1} : \textbf{q}\_{\text{eq}} = \frac{\textbf{q}\_{\text{max}} \textbf{K}\_{\text{RaP1}} \textbf{C}\_{\text{eq}}}{\left[\textbf{1} + \textbf{K}\_{\text{RP1}} \textbf{C}\_{\text{eq}}\right]^{\text{m}\_{\text{RaP1}}}} \tag{25}$$

$$\textbf{Model 2} : \textbf{q}\_{\text{eq}} = \frac{\textbf{q}\_{\text{max}} \textbf{K}\_{\text{RaP2}} \textbf{C}\_{\text{eq}}}{1 + \textbf{K}\_{\text{RP2}} \textbf{C}\_{\text{eq}}} \tag{26}$$

$$\textbf{Model 3} : \textbf{q}\_{\text{eq}} = \frac{\textbf{q}\_{\text{max}} \textbf{K}\_{\text{RaP3}} \textbf{C}\_{\text{eq}}^{\text{m}\_{\text{RaP3}}}}{\textbf{1} + \textbf{K}\_{\text{RP3}} \textbf{C}\_{\text{eq}}^{\text{m}\_{\text{RaP3}-1}}} \tag{27}$$

When both mRaP1 and mRaP2 are equal to 1, the Radke-Prausnitz 1, 2 models decrease to the Langmuir model; nevertheless, at low concentrations, the models become Henry's law; but, at high adsorbate concentrations, the Radke-Prausnitz 1 and 2 models become the Freundlich model. When the exponent mRaP3 is equal to 1, the Radke-Prausnitz-3 equation simplifies to Henry's law, and when the exponent mRaP3 is equal to 0, it becomes the Langmuir isotherm.

#### *3.3.7 Toth isotherm model*

The Toth adsorption isotherm model is used to explain heterogeneous adsorption systems that fulfil both the low and high end boundaries of adsorbate concentration [47]. Eq. (28) represents the Toth isotherm model.

$$\mathbf{q\_{eq}} = \frac{\mathbf{q\_{max}} \mathbf{C\_{eq}}}{\left(\frac{1}{\mathbf{K\_T}} + \mathbf{C\_{eq}^{n\_T}}\right)^{\frac{1}{m\_T}}} \tag{28}$$

When n = 1, this equation simplifies to the Langmuir isotherm equation, indicating that the process is approaching the homogenous surface. As a result, the value n describes the adsorption system's heterogeneity. The system is considered to be heterogeneous if it deviates farther from unity.

#### *3.3.8 Khan isotherm model*

For adsorption of bi-adsorbate from pure dilute equations solutions, the Kahn isotherm model is devised [48]. Eq. (29) presents the Kahn isotherm model.

$$\mathbf{q}\_{\rm eq} = \frac{\mathbf{q}\_{\rm m} \mathbf{b}\_{\rm K} \mathbf{C}\_{\rm eq}}{\left(\mathbf{1} + \mathbf{b}\_{\rm K} \mathbf{C}\_{\rm eq}\right)^{\rm a\rm c}} \tag{29}$$

When aK is equal to one, the Toth model approaches the Langmuir isotherm model, and when aK is more than one, the Toth model simplifies to the Freundlich isotherm model.

#### *3.3.9 Koble-Corrigan isotherm model*

The Sips isotherm model is similar to the Koble-Carrigan isotherm model. This model includes both the Langmuir and the Freundlich isotherms [49]. Eq. (30) depicts the Koble-Carrigan isotherm model.

$$\mathbf{q}\_{\rm eq} = \frac{\mathbf{A}\_{\rm KC} \mathbf{C}\_{\rm eq}^{\rm n\rm c}}{\mathbf{1} + \mathbf{B}\_{\rm KC} \mathbf{C}\_{\rm eq}^{\rm n\rm c}} \tag{30}$$

This model reduces to the Freundlich isotherm at large adsorbate concentrations. It is only acceptable when n is higher than or equal to 1. When n is lesser than one, it indicates that the model, despite a high concentration coefficient or a low error value, is incapable of describing the experimental data.

#### *3.3.10 Jossens isotherm model*

The Jossens isotherm model is based on the energy distribution of adsorbateadsorbent interactions at adsorption sites [50]. At low concentrations, this model is reduced to Henry's law. Eq. (29) depicts the Jossens isotherm model.

*Removal of Divalent Nickel from Aqueous Solution Using Blue Green Marine Algae… DOI: http://dx.doi.org/10.5772/intechopen.103940*

$$\mathbf{q}\_{\rm eq} = \frac{\mathbf{K}\_{\rm f} \mathbf{C}\_{\rm eq}}{\mathbf{1} + \mathbf{J} \begin{array}{c} \mathbf{C}\_{\rm eq} \\ \end{array}} \tag{31}$$

At low capacity, J equates to Henry's constant. bJ is the Jossens isotherm constant, which is constant regardless of temperature or the composition of the adsorbent.

#### *3.3.11 Jovanovic-Freundlich isotherm model*

To depict single-component adsorption equilibrium on heterogeneous surfaces, the Jovanovic-Freundlich isotherm model is developed [51]. Eq. (32) represents the Jovanovic-Freundlich isotherm model.

$$\mathbf{q}\_{\rm eq} = \mathbf{q}\_{\rm max} \left[ \mathbf{1} - \mathbf{e}^{-\left(\mathbf{K}\_{\rm IF} \ \mathbf{C}\_{\rm eq} \ \mathbf{"} \right)} \right] \tag{32}$$

#### *3.3.12 Brouers-Sotolongo isotherm model*

This isotherm is built in the form of a deformed exponential function for adsorption onto a heterogeneous surface [52]. Eq. (33) depicts the Brouers-Sotolongo model.

$$\mathbf{q}\_{\rm eq} = \mathbf{q}\_{\rm max} \left[ \mathbf{1} - \mathbf{e}^{\left( -\mathbb{K}\_{\rm BS} \mathbf{C}\_{\rm eq}^{a\_{\rm RS}} \right)} \right] \tag{33}$$

The parameter αBS is related with distribution of adsorption energy and the energy of heterogeneity of the adsorbent surfaces at the given temperature [53].

### *3.3.13 Vieth-Sladek isotherm model*

This model includes two independent parts for calculating transient adsorption diffusion rates in solid adsorbents [54]. Eq. (34) represents the Vieth-Sladek isotherm model.

$$\mathbf{q\_{eq}} = \mathbf{K\_{VS}}\mathbf{C\_{eq}} + \frac{\mathbf{q\_{max}}\beta\_{\rm VS}\mathbf{C\_{eq}}}{\mathbf{1} + \beta\_{\rm VS}\mathbf{C\_{eq}}} \tag{34}$$

#### *3.3.14 Unilan isotherm model*

The application of the local Langmuir isotherm and uniform energy distribution is assumed for the Unilan isotherm model [44]. Eq. (35) presents the Unilan isotherm model.

$$\mathbf{q}\_{\rm eq} = \frac{\mathbf{q}\_{\rm max}}{2\beta\_{\rm U}} \ln \left[ \frac{\mathbf{1} + \mathbf{K}\_{\rm U} \mathbf{C}\_{\rm eq} \mathbf{e}^{\beta\_{\rm U}}}{\mathbf{1} + \mathbf{K}\_{\rm U} \mathbf{C}\_{\rm eq} \mathbf{e}^{-\beta\_{\rm U}}} \right] \tag{35}$$

The higher the model exponent βU, the system is more heterogeneous. If βU is equal to 0, the Unilan isotherm model becomes the classical Langmuir model as the range of energy distribution is zero in this limit [50, 55, 56].

#### *3.3.15 Holl-Krich isotherm model*

The Langmuir Isotherm [57] is a version of the Holl-Krich Isotherm Model. The Freundlich isotherm is formed when the concentration of the solvent is low [22]. The Holl-Krich Isotherm Model may be seen in Eq. (36).

$$\mathbf{q\_{eq}} = \frac{\mathbf{q\_{max}} \mathbf{K\_{HK}} \mathbf{C\_{eq}^{n\_{HK}}}}{\mathbf{1} + \mathbf{K\_{HK}} \mathbf{C\_{eq}^{n\_{HK}}}} \tag{36}$$

#### *3.3.16 Langmuir-Jovanovic isotherm model*

This empirical model is the combined form of both Langmuir and Jovanovic isotherm [58]. The Langmuir-Jovanovic model is given in Eq. (37).

$$\mathbf{q}\_{\rm eq} = \frac{\mathbf{q}\_{\rm max} \mathbf{C}\_{\rm eq} \left[ \mathbf{1} - \mathbf{e}^{\left( \mathbf{K}\_{\rm l} \mathbf{C}\_{\rm eq}^{\rm NL} \right)} \right]}{\mathbf{1} + \mathbf{C}\_{\rm eq}} \tag{37}$$

#### **3.4 Four parameter models**

The four parameter models discussed in this study are Fritz-Schlunder-IV, Baudu, Weber-van Vliet and Marczewski-Jaroniec models.

#### *3.4.1 Fritz-Schlunder-IV isotherm model*

Fritz-Schlunder IV model is another model comprised of four-parameter with combine features of Langmuir-Freundlich isotherm [45]. The model is given in Eq. (38).

$$\mathbf{q\_{eq}} = \frac{\mathbf{A\_{FS\ $}C\_{eq}^{\mathbf{a\_{FS\$ }}}}}{\mathbf{1} + \mathbf{B\_{FS\ $}C\_{eq}^{\mathbf{0\_{FS\$ }}}}} \tag{38}$$

When the values of αFS5 and βFS5 are less than or equal to one, this isotherm is true. The Fritz-Schlunder-IV isotherm transforms into the Freundlich equation at high adsorbate concentrations. If both αFS5 and βFS5 are equal to one, the isotherm is reduced to the Langmuir isotherm. This isotherm model becomes the Freundlich at large concentrations of adsorbate in the liquid-phase.

#### *3.4.2 Baudu isotherm model*

The Baudu isotherm model was created in response to a disagreement in computing the Langmuir constant and coefficient from slope and tangent across a wide range of concentrations [59]. The Langmuir isotherm model has been modified into the Baudu isotherm model. Eq. (39) explains it.

$$\mathbf{q}\_{\rm eq} = \frac{\mathbf{q}\_{\rm max} \ \mathbf{b}\_{\rm o} \ \mathbf{C}\_{\rm eq}^{(1+\mathbf{x}+\mathbf{y})}}{\mathbf{1} + \mathbf{b}\_{\rm o} \mathbf{C}\_{\rm eq}^{(1+\mathbf{x})}} \tag{39}$$

*Removal of Divalent Nickel from Aqueous Solution Using Blue Green Marine Algae… DOI: http://dx.doi.org/10.5772/intechopen.103940*

This model is only applicable in the range of (1 + x + y) < 1 and (1 + x) < 1. For lower surface coverage, Baudu model reduces to the Freundlich equation [58], i.e.:

$$\mathbf{q}\_{\rm eq} = \frac{\mathbf{q}\_{\rm m0} \,\mathrm{b}\_{\rm o} \mathbf{C}\_{\rm eq}^{(\rm + \, x + y)}}{\mathbf{1} + \mathbf{b}\_0} \tag{40}$$

#### *3.4.3 Weber-van Vliet isotherm model*

With four parameters, the Weber and van Vliet isotherm model is used to represent equilibrium adsorption data [60–62]. Eq. (41) depicts the model.

$$\mathbf{C\_{eq}} = \mathbf{P\_1q\_{eq}}^{\left(\mathbf{p\_{2}q\_{eq}^{\mathbf{P\_3}} + p\_4}\right)} \tag{41}$$

Multiple nonlinear curve fitting approaches based on the reduction of the sum of squares of residuals can be used to define the isotherm parameters P1, P2, P3, and P4.

#### *3.4.4 Marczewski-Jaroniec isotherm model*

The Marczewski-Jaroniec isotherm model is analogous to the Langmuir isotherm model [62, 63]. Eq. (42) represents the Marczewski-Jaroniec isotherm model.

$$\mathbf{q}\_{\rm eq} = \mathbf{q}\_{\rm max} \left[ \frac{\left( \mathbf{K}\_{\rm MJ} \mathbf{C}\_{\rm eq} \right)^{\rm n\_{\rm MJ}}}{\mathbf{1} + \left( \mathbf{K}\_{\rm MJ} \mathbf{C}\_{\rm eq} \right)^{\rm n\_{\rm MJ}}} \right] \tag{42}$$

The spreading of distribution along the route of increasing adsorption energy is described by KMJ.

#### **3.5 Five parameter model**

Accounting for the high parameter models offers unmistakable information on the process of adsorption under equilibrium conditions. Only one five-parameter model, the Fritz-Schlunder-V isotherm model, is used in this section.

### *3.5.1 Fritz-Schlunder-V isotherm model*

The Fritz-Schlunder adsorption isotherm model was created with the goal of more precisely reproducing model modifications for applicability over a wide range of equilibrium data [45]. Eq. (43) represents the Fritz-Schlunder adsorption isotherm model.

$$\mathbf{q}\_{\rm eq} = \frac{\mathbf{q}\_{\rm max} \mathbf{K}\_{\rm IFSS} \mathbf{C}\_{\rm eq}^{\rm aggs}}{\mathbf{1} + \mathbf{K}\_{\rm zFSS} \mathbf{C}\_{\rm eq}^{\beta\_{\rm FS}}} \tag{43}$$


**1.6 Principle of maximum entropy production and oceanic general circulation**

In this sub-section, we briefly explain another important thermodynamic postulate of stability of a nonlinear non-equilibrium system such as the oceanic general circulation, the principle of the maximum Entropy Production and consider the stability of oceanic general circulation from a global perspective because local processes of generation and dissipation of kinetic energy in a turbulent medium remain unknown.

The ocean system can be regarded as an open non-equilibrium system connected with surrounding systems mainly via heat and salt fluxes. The surrounding systems consist of the atmosphere, the Sun and space. Because of the curvature of the Earth's surface and the inclination of its rotation axis relative to the Sun, net gains of heat and salt are found in the equatorial region; net losses of heat and salt are apparent in polar regions. The heat and salt fluxes bring about an inhomogeneous distribution of temperature and salinity in the ocean system. This inhomogeneity produces the circulation, which in turn reduces the inhomogeneity. In this respect, the formation of the circulation can be regarded as a process leading to final equilibrium of the whole system: the ocean system and its surroundings. In this process, the rate of approach to equilibrium, i.e., the rate of entropy production by the oceanic circulation, is an important factor.

Related to the rate of entropy production in an open non-equilibrium system, Sawada (1981) reported that such a system tends to follow a path of evolution with a maximum rate of entropy production among manifold dynamically possible paths. This postulate has been called the principle of Maximum Entropy Production (MEP), which has been confirmed as valid for mean states of various nonlinear fluid systems, e.g., the global climate system of the Earth (Ozawa & Ohmura, 1997; Paltridge, 1975, 1978), those of other planets (Lorenz et al., 2001), the oceanic general circulation including both surface and abyssal circulations (Shimokawa, 2002; Shimokawa & Ozawa, 2001, 2002, 2007), and thermal convection and shear turbulence (Ozawa et al., 2001). Therefore, it would seem that MEP can stand for a

Thermodynamics of the Oceanic General Circulation –

Is the Abyssal Circulation a Heat Engine or a Mechanical Pump? 153

Fig. 3. (a) Model domain, and forcing fields of the model as functions of latitude, (b) forced

Fig. 4. The zonally integrated meridional stream function at years (a) 100, (b) 1000, (d) 2000, (e) 3000, (d) 4000, and (e) 5000 after starting the numerical calculations. The contour line interval is 2 SV (106 m3 s-1). The circulation pattern reached a statistically steady-state after

zonal wind stress (N m-2) defined as positive eastward, (c) prescribed sea surface

temperature (oC), and (d) prescribed sea surface salinity (‰).

year 4000.

universal principle for time evolution of non-equilibrium systems (see reviews of Kleidon and Lorenz, 2005; Lorenz, 2003; Martyushev & Seleznev, 2006; Ozawa et al., 2003; Whitfield, 2005). However, although some attempts have been made to seek a theoretical framework of MEP (e.g., Dewar, 2003, 2005), we remain uncertain about its physical meaning.


**CURRENT TOPICS IN GASTRITIS - 2012**

Edited by **Gyula Mózsik**

#### **Current Topics in Gastritis - 2012**

http://dx.doi.org/10.5772/46056 Edited by Gyula Mozsik

#### **Contributors**

Alejandro H. Corvalan, Gonzalo Carrasco, Kathleen Saavedra, Bruna Maria Roesler, José Murilo Zeitune, Shotaro Enomoto, Kiron Moy Das, Jiro Watari, Kentaro Moriichi, Hiroki Tanabe, Mikihiro Fujiya, Hiroto Miwa, Yutaka Kohgo, Gonzalo Castillo-Rojas, German Aguilar- Gutierrez, Eduardo Mucito-Varela, Stephanie E. Morales-Guerrero, Yolanda López-Vidal, Mohamed Elseweidy, Hanan Alshenawy, Zheming Lu, Dajun Deng, Gyula Mozsik, Jozsef Czimmer, Imre Laszlo Szabo, Janos Szolcsanyi, Kata Cseko, Paolo Arcari, S. Swarnakar, Achariya - Sailasuta

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**1. Introduction**

Because it is often difficult to treat central nervous system (CNS) disorders using standard pharmacological methods, other, more effective, strategies are being sought. Among these alternatives, gene therapy appears to be a promising approach to treating various neurological disorders, including Parkinson's disease, amyotrophic lateral sclerosis, Huntington disease, Alzheimer's disease, and lysosomal storage disorders, which are not treatable by any other methods. However, the efficacy of gene therapies in clinical trials has been limited by phys‐ iological barriers unique to the CNS, as well as by the post-mitotic state of many of the cellular targets in the brain and spinal cord. The blood–brain barrier (BBB) in particular is a key obstacle to gene delivery to the CNS.

© 2015 The Author(s). Licensee InTech. This chapter is 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.

Adeno-associated virus (AAV) is a member of the family Parvoviridae that has been widely used as a vector for gene delivery. AAV is a small, non-enveloped single-stranded DNA virus with a genome of approximately 4.7 kb [1]. The AAV genome consists of three open reading frames (ORFs) flanked by two inverted terminal repeats (ITRs). It is a dependovirus because it requires helper functions from other viruses such as adenovirus or herpes simplex virus for its replication. AAV is a suitable gene transfer tool because of its safety due to a lack of pathogenicity, its ability to transduce both dividing and non-dividing cells, and its minor immune response. Among the more than 100 nonredundant AAV genotypes that have been identified, 12 AAV serotypes with unique properties have been used to produce most expression vectors [2].

AAV vectors are powerful tools that are able to mediate gene transfer to the CNS, thanks to their safety and ability to transduce non-dividing neuronal cells. Consequently, they hold great potential for use in therapeutic gene delivery strategies for the treatment of neurological disorders. Although a breakthrough treatment has remained elusive, current approaches are now considerably safer and potentially much more effective than in the past. In this chapter, we discuss how to administer vectors across the BBB, focusing especially on potential routes for administration of AAV vectors and promising strategies for application of AAV vectors in CNS disorders.


**Antimicrobial Resistance and Anti-Biofilms**

Editors

**Junyan Liu Ding-Qiang Chen Yulong Tan Ren-You Gan Guanggang Qu Zhenbo Xu**

MDPI ' Basel ' Beijing ' Wuhan ' Barcelona ' Belgrade ' Manchester ' Tokyo ' Cluj ' Tianjin

*Editorial Office* MDPI St. Alban-Anlage 66 4052 Basel, Switzerland

This is a reprint of articles from the Special Issue published online in the open access journal *Antibiotics* (ISSN 2079-6382) (available at: www.mdpi.com/journal/antibiotics/special issues/anti-Biofilms).

For citation purposes, cite each article independently as indicated on the article page online and as indicated below:

LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. *Journal Name* **Year**, *Volume Number*, Page Range.

**ISBN 978-3-0365-6951-2 (Hbk) ISBN 978-3-0365-6950-5 (PDF)**

© 2023 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications.

The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND.


**6. Future perspectives**

At least 50% of periodontitis vulnerability is attributed to heredity or genetic factors [20]. Clinical observations and scientific studies have demonstrated that the heredity of a host response pattern may be an important susceptibility factor in developing periodontal diseases [20, 79, 80]. Added information from new technologies, such as micro-arrays and DNA-sequencing, lead to the identification of specific genetic, environmental, and behavioural factors that influence periodontitis susceptibility [81].

In order to enhance the therapeutic management of periodontal disease, we must not only be able to identify genetic determinants, but also learn to modify, control or modulate the host response either by stimulating a desired immune response, or by decreasing the activating factors of bone resorption,both of which hinders the progression of the disease by [82–84].


*Necrotizing Enterocolitis DOI: http://dx.doi.org/10.5772/intechopen.85784*

*Pediatric Surgery, Flowcharts and Clinical Algorithms*

except in term babies [21].

isolated in 3% of cases.

**4. Pathology**

abstinence; fetal growth restriction; and perinatal hypoxia [20]. Despite the fact that no predilection for sex, race, or ethnicity has been conclusively established, a higher incidence is observed in male African American infants than in any other single demographic. This could be related to the higher incidence of prematurity in this ethnic group than in the general US population. Hypoxic ischemic injury is no longer considered a major predisposing factor in the development of NEC

NEC primarily affects the ileum and colon, the commonest location being ileocecal area [22]. The entire gastrointestinal tract may be involved in severe cases. On gross examination, the bowel loops are distended with areas of hemorrhage, congestion, necrosis, and pneumatosis (**Figure 1**). On microscopic examination, signs of inflammation, mucosal edema, bacterial invasion, submucosal and intramural gas bubbles, and ischemic transmural necrosis are seen. Intestinal perforation may happen when the entire thickness of bowel is involved leading to pneumoperitoneum, peritonitis, and portal venous gas (**Figure 2**). Microscopically, the predominant feature is coagulation necrosis, suggesting an ischemic origin of NEC. The aggregated inflammatory cells are both acute and chronic, such as neutrophils, lymphocytes, and macrophages representing an appropriate response to pathogenic bacterial invasion and tissue necrosis. Epithelial regeneration, granulation tissue formation and fibrosis may be seen suggesting reparative histological process [23]. Common pathogens isolated in NEC are Enterobacteriaceae including *Escherichia*, *Salmonella*, *Enterobacter*, and *Klebsiella* (68%); staphylococcal species (26%); clostridium species (4%); viruses including rota, echo, corona, and toro (11%); and candida (1%). No organism is

*Macroscopic appearance of necrotizing enterocolitis showing necrotic bowel loops. [Courtesy of Renu Sharma,*

*MD, Professor of Pediatrics, University of Florida at Jacksonville, USA].*

**28**

**Figure 1.**

**Figure 2.** *Schematic presentation of pathogenesis of NEC.*


**Types of Optical Gas Sensors**

#### *Optical Gas Sensors DOI: http://dx.doi.org/10.5772/intechopen.108971*

A waveguide with a fiber core and surrounding cladding makes up its primary part. The core allows a small amount of light to transmit in form of evanescent waves perpendicular to the axis of fiber with exponentially diminished intensity in the cladding. On the other hand, majority of the light travels along with the fiber core via total internal reflection because core has a greater refractive index than the cladding. The alteration in the refractive index of the cladding is caused by the adsorption of the molecules of gas on its surface, which in turn affects the wavelength and intensity of the output light signals. Due to the limited selectivity of the gas adsorptioninduced variation in the overall refractive index of the material, the cladding is typically modified with molecules or nano-materials that have a particular affinity for the target gas.


**2. Model Predictive Control**

Model predictive control is a form of control scheme in which the current control action is obtained by solving, at each sampling instant, a finite horizon open-loop optimal control problem, using the current state of the plant as the initial state. Then the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant. This is the main difference from conventional control which uses a pre-computed control law.

#### **2.1. Characteristics of MPC**

MPC is by now a mature technology. It is fair to mention that it is the standard approach for implementing constrained, multivariable control in the process industries today. Further‐ more, MPC can handle control problems where off-line computation of a control law is diffi‐ cult or impossible.

Specifically, an important characteristic of this type of control is its ability to cope with hard constraints on controls and states. Nearly every application imposes constraints. For in‐ stance, actuators are naturally limited in the force (or equivalent) they can apply, safety lim‐ its states such as temperature, pressure and velocity, and efficiency, which often dictates steady-state operation close to the boundary of the set of permissible states. In this regard, MPC is one of few methods having applied utility, and this fact makes it an important tool for the control engineer, particularly in the process industries where plants being controlled are sufficiently slow to permit its implementation.

In addition, another important characteristic of MPC is its ability to handle control problems where off-line computation of a control law is difficult or impossible. Examples where MPC may be advantageously employed include unconstrained nonlinear plants, for which online computation of a control law usually requires the plant dynamics to possess a special structure, and time-varying plants.

A fairly complete discussion of several design techniques based on MPC and their relative merits and demerits can be found in the review article by [4].

#### **2.2. Essence of MPC**

dictive control (RMPC) techniques have been developed in recent decades. We review dif‐ ferent RMPC methods which are employed widely and mention the advantages and disadvantages of these methods. The basic idea of each method and some method applica‐

Most MPC strategies consider hard constraints, and a number of RMPC techniques exist to handle uncertainty. However model and measurement uncertainties are often stochastic, and therefore RMPC can be conservative since it ignores information on the probabilistic distribution of the uncertainty. It is possible to adopt a stochastic uncertainty description (in‐ stead of a set-based description) and develop a stochastic MPC (SMPC) algorithm. Some of

In recent years, there has been much interest in networked control systems (NCSs), that is, control systems close via possibly shared communication links with delay/bandwidth con‐ straints. The main advantages of NCSs are low cost, simple installation and maintenance, and potentially high reliability. However, the use of the network will lead to intermittent losses or delays of the communicated information. These losses will tend to deteriorate the performance and may even cause the system to become unstable. MPC framework is partic‐ ularly appropriate for controlling systems subject to data losses because the actuator can profit from the predicted evolution of the system. In section 7, results from our recent re‐ search are summarized. We propose a new networked control scheme, which can overcome

At the beginning of research on NCSs, more attention was paid on single plant through net‐ work. Recently, fruitful research results on multi-plant, especially, on multi-agent net‐ worked control systems have been obtained. MPC lends itself as a natural control framework to deal with the design of coordinated, distributed control systems because it can account for the action of other actuators while computing the control action of a given set of actuators in real-time. In section 8, a number of distributed control architectures for inter‐ connected systems are reviewed. Attention is focused on the design approaches based on

Model predictive control is a form of control scheme in which the current control action is obtained by solving, at each sampling instant, a finite horizon open-loop optimal control problem, using the current state of the plant as the initial state. Then the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant. This is the main difference from conventional control which uses a pre-computed control law.

MPC is by now a mature technology. It is fair to mention that it is the standard approach for implementing constrained, multivariable control in the process industries today. Further‐

tions are stated as well.

78 Advances in Discrete Time Systems

the recent advances in this area are reviewed.

the effects caused by the network delay.

model predictive control.

**2.1. Characteristics of MPC**

**2. Model Predictive Control**

As mentioned in the excellent review article [2], MPC is not a new method of control design. Rather, it essentially solves standard optimal control problems which is required to have a finite horizon in contrast to the infinite horizon usually employed in *H*2and *H∞*linear opti‐ mal control. Where it differs from other controllers is that it solves the optimal control prob‐ lem on-line for the current state of the plant, rather than providing the optimal control for all states, that is determining a feedback policy on-line.

The on-line solution is to solve an open-loop optimal control problem where the initial state is the current state of the system being controlled which is just a mathematical programming problem. However, determining the feedback solution, requires solution of Hamilton-Jaco‐ bi-Bellman (Dynamic Programming) differential or difference equation, a vastly more diffi‐ cult task (except in those cases, such as *H*2and *H*∞linear optimal control, where the value function can be finitely parameterized). From this point of view, MPC differs from other control methods merely in its implementation. The requirement that the open-loop optimal control problem be solvable in a reasonable time (compared with plant dynamics) necessi‐ tates, however, the use of a finite horizon and this raises interesting problems.


# **Decline of Mediterranean Fruit Crops and Forests Associated with Fungal Trunk Pathogens**

Edited by Carlos Agustí-Brisach Printed Edition of the Special Issue published in *Plants*

www.mdpi.com/journal/plants


**1. Introduction**

It could be thought, initially, that this chapter will be one more that deals with the impact that certain invasive species cause on the biota of a certain area, region, or country. Not exactly, though too. It is necessary to recognize that this matter - much approached and treated at different levels - still leaves ample room for suggestions and debates. In the text that follows, several examples of intervention in favor of a specific bird species are shown, to increase its populations and "remove" it from a classification "critically endangered" or remarkably close to it.

The measures implemented by different public administrations and/or conservation entities, to try to recover a specific species and avoid its classification as endangered, near endangered, etc., have provoked, and still do, a series of reactions both target species in question, such as related species or prey species; even on those that we would never have believed could be affected by the recovery plans of the former.

Admittedly, some of these reactions were not easy to predict, others were not. In some cases, political or personal "honor" criteria have prevailed, in an apparent maelstrom or race, to have more individuals than another "competing" region or nation.

Also, shortcuts have sometimes been sought. Some original, others of questionable ethics, with reasonable doubts to be protected by scientific criteria.

At times, there has been a sin of precipitation and rapid search for striking results, rather than the application of biological knowledge, and of considering a probable long-term projection.

"A wild population is defined as a set of individuals of a species that inhabits a certain area". "Density is the number of specimens per spatial unit (surface or volume). It is often more useful than the absolute size of the population since density determines and conditions fundamental aspects such as competition for resources" [1].

"Traditionally, indicator species have been considered those that, by being present in a certain system, indicate that said ecosystem is healthy, from the physical, chemical, or biological point of view (or, by the on the contrary, that it is deteriorated, as occurs with the species of aquatic invertebrates that indicate contamination). They are usually species that are easy to detect and "monitor" so that the demographic changes of their populations can be detected in time and interpreted in terms of other variables of conservation interest that are more difficult to measure" [2].

Dedicating all efforts to the recovery of a species, you can avoid the symptoms that warn us of its risk situation, but not end with its origin or, what is worse, camouflage the situation of other less striking species, but in a similar situation. "No species serves to indicate anything about its ecosystem of origin when it is kept in captivity, or conditions of probation, away from enemies and provided with food" [2].

Recovery involves much more than just increasing population numbers. Density can and should have a limit.

These premises constitute the basis on which this chapter will work: The growth in the number of specimens/surface unit of a specific species, which has been "helped" in different ways, is affecting other populations in such a way that it has displaced or eradicated them from specific areas, endangering their existence.

Not all cases of increase in population density in birds are due to human intervention, through the execution of direct intervention measures, to intentionally favor a certain endangered species. However, behind most animal overpopulations - not just birds - are human actions that, consciously or unconsciously, have caused them.

**93**

*The Limit to the Density of Species (A Reflection on Human Intervention in Conservation...*

The examples discussed below refer to situations that occurred in Spain. We know of multiple similar actions in other countries. The reader will be able to relate and apply each case to its environment but will agree with us that a limit to the

Traditionally, raptors have been considered "harmful" in a simplistic dichotomous classification, in contrast to "beneficial birds." The diurnal raptors have been especially "persecuted", for predating on rabbits (*Oryctolagus cuniculus),* red-legged partridges (*Alectoris rufa*), and other species, which man uses for his consumption. Birds of prey were considered protected species, in Spain, from the year 1973. However, their habitats were not protected until several years later. It could be the paradox that an imperial eagle (*Aquila adalberti*) would fly over the entire Spanish territory without risk of being shot, but, incoherently, it would starve because its habitat had been destroyed, it did not have preys, or it could not rest safely or

The custom of poisoning carrion and eggs did not cease with the law. Raptors continued to perish due to the "silent" effect of strychnine, a situation that also affected corvids. The secondary poisoning of scavenger birds occurred by ingesting the corpses of mammals considered vermin, such as foxes, which had been poisoned, or even wild boars, which had eaten the deadly bait intended for foxes. There was another important decrease in the populations of griffon vultures due to the use of the tractor in agriculture, because it marked the end of the use of animals to plow and harvest agricultural fields: animals that were a basic contribu-

Traditionally, the corpses of domestic cattle were abandoned in the field, in the same place where they died (in the case of extensive livestock) or transferred to a

In the case of the herds in the extensive regime, the location of the corpses depended completely on chance, so the ethological patterns of the search flights of

The European Union, with the emergence of the bovine disease, called spongiform encephalopathy, which is transmissible to humans, issued regulations that prohibited these practices, making it mandatory to hire incineration services for carcasses. Suddenly, the most important source of food for scavengers was eliminated, which from then on only had carcasses from hunting. In some localities, points of deposit of carcasses of cattle were authorized, in which the animals were

During that time, vultures attacked disadvantaged animals, such as during parturition, tearing of genitalia and killing the calf or lamb [5]. Situations that, until then, had been exceptional, but continues occurring on more frequent occasions today (**Figure 1**). As a contrast [3–5], as follows: "A sheep of my property died in childbirth, I dragged it to a clearing where the vultures could see it better, they fell on it and almost ate it completely but they did not touch the corpse of the dead lamb as well and cause the sheep to die. It is the case of a dead animal" [4] and "A cow disappeared from a farm in Portezuelo, a town very close to Acehúche. It was found almost eaten by vultures and suddenly we saw something moves on the corpse; it was a calf that had not been touched and left ahead. In this case they were able to

reproduce, because there are no trees on which to build the nest.

point away from the facilities if they were housed animals.

eviscerated (being buried or incinerated these carcasses).

attack the mother even before she was dead, but we are not sure" [4].

*DOI: http://dx.doi.org/10.5772/intechopen.97436*

density of the species is necessary.

**2.1 The griffon vulture (***Gyps fulvus***)**

**2. The case of …**

tion to their subsistence.

the vultures did not suffer alterations.

*The Limit to the Density of Species (A Reflection on Human Intervention in Conservation... DOI: http://dx.doi.org/10.5772/intechopen.97436*

The examples discussed below refer to situations that occurred in Spain. We know of multiple similar actions in other countries. The reader will be able to relate and apply each case to its environment but will agree with us that a limit to the density of the species is necessary.