# Modelling and analysis of a modified May–Holling–Tanner predator-prey model
with Allee effect in the prey and an alternative food source for the predator

Claudio Arancibia–Ibarra
School of Mathematical Sciences, Queensland University of Technology
GPO Box 2434, GP Campus, Brisbane, Queensland 4001 Australia
Facultad de Ingeniería y Negocios, Universidad de Las Américas
Av. Manuel Montt 948, Santiago, Chile
<EMAIL_ADDRESS>
José Flores
Department of Mathematics, The University of South Dakota (USD)
Vermillion, South Dakota, USA
<EMAIL_ADDRESS>

###### Abstract

In the present study, we have modified the traditional May–Holling–Tanner
predator-prey model used to represent the interaction between least weasel and
field-vole population by adding an Allee effect (strong and weak) on the
field-vole population and alternative food source for the weasel population.
It is shown that the dynamic is different from the original May–Holling–Tanner
predator-prey interaction since new equilibrium points have appeared in the
first quadrant. Moreover, the modified model allows the extinction of both
species when the Allee effect (strong and weak) on the prey is included, while
the inclusion of the alternative food source for the predator shows that the
system can support the coexistence of the populations, extinction of the prey
and coexistence and oscillation of the populations at the same time.
Furthermore, we use numerical simulations to illustrate the impact that
changing the predation rate and the predator intrinsic growth rate have on the
basin of attraction of the stable equilibrium point or stable limit cycle in
the first quadrant. These simulations show the stabilisation of predator and
prey populations and/or the oscillation of these two species over time.

_K_ eywords Modified May–Holling–Tanner $\cdot$ Holling–Tanner type II $\cdot$
Alternative food $\cdot$ Allee effect.

##  1 Introduction

Numerous investigations have extensively studied the interactions between the
specialist predator least-weasel (Mustela nivalis) and the field-vole
(Microtus agrestis) [hanski2, hanski, turchin2]. These investigations have
revealed important findings in the variation of the populations in
Fennoscandia from regular cycles to stable populations. This variation is also
affected by the fact that the rodent breeding is seasonal, and the duration of
the breeding varies with the physical location in Fennoscandia [dalkvist].
Moreover, cycles may be affected by the existence of an optimum group size and
an Allee effect [madison]. Graham and Lambin [graham] showed that field-vole
survival depends on the variation of weasel predation. They also demonstrated
that the proportion of weasels was suppressed in summer and autumn, while the
vole population always declined to a low density. Thus, if the predation by
weasels is reduced then the field-vole survival increases. In [graham], the
authors assumed that the May–Holling–Tanner model was not appropriate to
conduct such studies due to the large number of parameters. Additionally, the
authors suggested that there are different variations in these two species
that should be considered in order to improve study the results.

The most common mathematical framework for describing this predator-prey
interaction is that the weasel and field-vole have a logistic-type growth
function and the predator carrying capacity is prey dependent [turchin]. The
classic model used to represent the predation between these two species is the
May–Holling–Tanner predator-prey model [may] which is a modification of
original the Leslie–Gower model. Leslie (1948) proposed the following
generalisation of the Lotka-Volterra model

$\displaystyle\dfrac{dN}{dt}$
$\displaystyle=rN\left(1-\dfrac{N}{K}\right)-qNP,$ (1)
$\displaystyle\dfrac{dP}{dt}$ $\displaystyle=sP\left(1-\dfrac{P}{nN}\right).$

where $N(t)$ is used to represent the size of the prey population at time $t$,
$P(t)$ is used to represent the size of the predator population at time $t$,
$r$ is the intrinsic growth rate for the prey, $s$ is the intrinsic growth
rate for the predator, $K$ is the prey environmental carrying capacity, $n$ is
a measure of the quality of the prey as food for the predator, $nN$ can be
interpreted as a prey dependent carrying capacity for the predator and the
term $P/nN$ is known as the Leslie-Gower term. It measures the loss in the
predator population due to rarity (per capita $P/N$) of its favourite food.

This model retains the prey equation from the Volterra model, but for the
predator it assummes a logistic-like term, in which the predator carrying
capacity is directly proportional to prey density. This interesting
formulation for the predator dynamics has been discussed in Leslie and Gower
in [leslie] and Pielou in [pielou]. Later, May (1974) in [may] modified the
Leslie-Gower model (1) by assuming the hyperbolic functional response (Holling
type II) in the prey equation, $H(N)=qN/(N+a)$, which occurs in species when
the number of prey consumed rises rapidly at the same time that the prey
density increases [turchin, may, santos]. The parameter $q$ is the maximum
predation rate per capita and $a$ is half of the saturated level [turchin]. In
particiular, it is given by

$\displaystyle\dfrac{dN}{dt}$
$\displaystyle=rN\left(1-\dfrac{N}{K}\right)-q\dfrac{NP}{N+a}.$ (2)
$\displaystyle\dfrac{dP}{dt}$ $\displaystyle=sP\left(1-\dfrac{P}{nN}\right).$

Model (2) is known as Holling–Tanner or as May–Holling–Tanner model which is
described by an autonomous two-dimensional system of ordinary differential
equations [turchin, aguirre1, arancibia7].

The original May–Holling–Tanner model (2) was studied, among others, by
[saez]. The authors proved that all the solutions which are initiated in
$\mathbb{R}^{2}_{+}$ are bounded and the equilibrium point $(1,0)$, which is
the scaled equilibrium point $(K,0)$, is a saddle point. Moreover, by doing a
polar blowing-up [dumortier] at the origin, Saez and González-Olivares [saez]
showed that there exists a separatrix curve between a hyperbolic and a
parabolic sector in the neighbourhood of the origin. The authors also found
that there is always one positive equilibrium point in the first quadrant
which can be stable; unstable surrounded by a stable limit cycle; or stable
surrounded by two limit cycles. Additionally, the global stability of a
periodic solution of this predator-prey model was investigated by [lisena] and
the analysis of a discrete May–Holling–Tanner model was described in [cao].

Figure 1: We show the per capita growth rate of the logistic function (blue
line), the strong Allee effect with $m=15$ (red curve), and the weak Allee
effect with $m=-15$ (orange curve) and $m=0$ (grey curve).

Additional complexity can be incorporated into the predator-prey model (2) in
order to make it more realistic. We can consider, in the prey population, a
mechanism connected with a density-dependent phenomenon (Allee effect [allee2,
allee, berec, buffoni, van, wang7]). The Allee effect may appear due to a wide
range of biological phenomena such as reduced anti-predator vigilance; social
thermo-regulation; genetic drift; mating difficulty; reduced defence against
the predator; and deficient feeding because of low population densities
[stephens2]. The Allee effect may appear in the field-vole population due to
the density-dependent habitat selection showed in [jankovic, morris].
Moreover, the stabilisation of field-vole populations depends on the variation
in reproductive rate and recruitment of the population [ostfeld]. The
influence of the Allee effect upon the logistic-type growth in the prey
equation is represented by the inclusion of a multiplier in the form of $N-m$
(3). For $0<m<K$, the per-capita growth rate of the prey population with the
Allee effect included is negative, but increasing, for $N\in[0,m)$, and this
is referred to as the “strong Allee effect”. When $m\leq 0$111Note that $m=0$
is often also called the “weak Allee effect”, however, the behaviour of the
model with $m=0$ is similar to the case of a strong Allee effect ($m>0$) and
thus we will not consider the case of $m=0$ in this manuscript., the per-
capita growth rate is positive but increases at low prey population densities
and this is referred to as the weak Allee effect [berec, courchamp2]. The per
capita growth rate for the logistic growth function and Allee effects (strong
and weak) are shown in Figure 1. We observe that the per capita growth rate
for a weak Allee effect is positive and growing, when compared to the strong
Allee effect. Moreover, the trajectories of a population affected by a weak
Allee effect are delayed compared with a strong Allee effect. This effect can
be generated by the reduction of the male population [liermann] or the habitat
selection [jankovic, morris].

Table 1: In the first row the prey growth rate is affected by Allee effect and in the second row the predator growth rate is affected by an alternative food source. Modified growth equations | Modified models
---|---
Allee effect: |
$F_{m}(N)=r\left(1-\dfrac{N}{K}\right)(N-m)$ (3) |  $\displaystyle\dfrac{dN}{dt}$ $\displaystyle=rN\left(1-\dfrac{N}{K}\right)\left(N-m\right)-\dfrac{qNP}{N+a},$ (4) $\displaystyle\dfrac{dP}{dt}$ $\displaystyle=sP\left(1-\dfrac{P}{nN}\right).$ (4)
Alternative food: |
$G_{m}(N,P)=s\left(1-\dfrac{P}{nN+c}\right)$ (5) |  $\displaystyle\dfrac{dN}{dt}$ $\displaystyle=rN\left(1-\dfrac{N}{K}\right)-\dfrac{qNP}{N+a},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ (6) $\displaystyle\dfrac{dP}{dt}$ $\displaystyle=sP\left(1-\dfrac{P}{nN+c}\right).$ (6)
Allee effect and alternative food: |
equations (3) and (5) |  $\displaystyle\dfrac{dN}{dt}$ $\displaystyle=rN\left(1-\dfrac{N}{K}\right)\left(N-m\right)-\dfrac{qNP}{N+a},$ (7) $\displaystyle\dfrac{dP}{dt}$ $\displaystyle=sP\left(1-\dfrac{P}{nN+c}\right).$ (7)

Some species are considered as a generalist and specialist predator and thus
it can switch to another available food, although its population growth may
still be limited by the fact that its preferred food is not available in
abundance. For instance, McDonald et al.  [mcdonald] reveal that weasels’ diet
involved mainly field-vole, which correspond to $68\%$ of the diet which can
be considered as the primary food source. Moreover, the authors also reveal
that lagomorphs with $25\%$ of the diet; birds and birds’ eggs with $5\%$ of
the diet and other types of food with $2\%$ of the diet can be considered as a
secondary food source, see Figure 2. This characteristic was not considered in
investigations associated with the interaction between the field-vole as a
prey and least-weasel as a predator [hanski2, hanski, turchin, wollkind,
hanski3, andersson, erlinge, hansson]. That is, these studies do not consider
that, since the predator is a generalist, it can survive under different
environments and utilise a large range of food resources. Instead of adding
more species to the model, we assume that these other food sources are
abundantly available [mcdonald] and model this characteristic by adding a
positive constant $c$ to the environmental carrying capacity for the predator
[aziz]. Therefore, we have made a modification to the prey-dependent logistic
growth term in the predator equation, namely $nN$ is replaced by $nN+c$ (5).
As a result, we observe that the modified May–Holling–Tanner model with an
alternative food source (6) is no longer singular for $N=0$.

Figure 2: Main and secondary diet of a specialist predator least weasel
(mustela nivalis) studied in [mcdonald].

The aim of this manuscript is to extends the properties of the weasel and
field-vole interaction studied in [graham] by showing the impact of an Allee
effect on the prey (4) and an alternative food source for the predator (6) in
the original May–Holling–Tanner model. In addition, the analysis of the
modified model with weak Allee effect (i.e. systems (4) with $m<0$) and the
modified model considering an alternative food for the predator (i.e. system
(6)) show a strong connection with the findings presented in [turchin2,
hanski3]. Study results suggest that the system does not lose the coexistence
and oscillation of the population when a constant alternative food source for
weasels and a weak Allee effect on the prey are included. Moreover, the
survival of the field-vole does not depend on the level of predation when a
weak Allee on the prey is included [graham]. However, we have shown that the
May–Holling–Tanner model with strong Allee effect on the prey (i.e. systems
(4) with $m>0$) reduce the complexity of the dynamics in the modified model.
The modified model with strong Allee effect supports, depending on the initial
conditions, extinction and coexistence of both the prey and the predator
species.

The mathematical interaction in systems (2), considering the modifications
separately and together, is study further in Section 2. In Section 3, we
conclude the manuscript, presenting the impact of the modification on the
dynamic of the modified May–Holling–Tanner models. We also discuss the
ecological implications of the inclusion of the strong and weak Allee effects
on the prey and of an alternative food for the predator in a real model
studied in [turchin2, graham, hanski3].

##  2 Models

The results derived in [saez] are for the nondimensionalised versions of the
modified versions of systems (2) which is given by

$\displaystyle\dfrac{dN}{dt}$
$\displaystyle=rN\left(1-\dfrac{N}{K}\right)-\dfrac{qNP}{N+a},$
$\displaystyle\dfrac{dP}{dt}$ $\displaystyle=sP\left(1-\dfrac{P}{nN}\right).$

However, this nondimensionalised version is topologically equivalent to the
original version of the model as we used a diffeomorphism preserving the
orientation of time. Therefore, the results can be transferred to the original
May–Holling–Tanner model. Moreover, we can extend the analysis by using the
system parameters presented in [turchin2]. In this paper, Turchin and Hanski
used the original May–Holling–Tanner model and the authors showed that the
oscillation of the field-vole is driven by their interaction with the weasel.
This oscillation is obtained by using the system parameters presented in Table
2.

Table 2: Description of the system parameters, their units and values that will be used in this manuscript based in Turchin & Hanski [turchin2] results. We add an alternative food for the predator and the Allee threshold values. Description of parameter | Symbols | Values | Units
---|---|---|---
Intrinsic growth rate of field-vole population | $r$ | $4$ | yr-1
Field-vole carrying capacity | $K$ | $150$ | voles ha-1
Maximum consumption per weasel | $q$ | $500-700$ | voles yr-1 weasel-1
Half of the saturated level | $a$ | $6$ | voles ha-1
Intrinsic growth rate of weasel population | $s$ | $1-1.5$ | yr-1
Quality of the field-vole as food for the weasel | $n$ | $0.025$ | voles -1 weasel
Quality of alternative species as food for the weasel | $c$ | $0.01$ | weasel
Allee threshold | $m$ | $\pm 15$ | voles ha-1

Nowadays, there are several computational methods to find different type of
bifurcations. These methods are implemented in software packages such as
MATCONT [matcont2]. Figure 3 illustrates the Hopf bifurcation which was
detected with MATCONT for the parameters $r$, $K$, $a$, $n$ and $c$ fixed,
while we will vary $q$ and $s$, see Table 2.

###  2.1 Original model

The original May–Holling–Tanner predator-prey model with functional response
Holling type II (2) was studied mathematically, for instance, in [saez, huang]
and used to model real predator-prey interactions [hanski2, hanski, roux,
turchin]. The bifurcation diagram of the original May–Holling–Tanner model (2)
created with MATCONT [matcont2] and two typical phase portraits are shown in
the first row of Figure 3. The Hopf curve222The Hopf curve is where the
stability of an equilibrium point switches and, in some cases, a periodic
solution arises. for the May–Holling–Tanner model shows that if the predation
rate $q$ and the intrinsic growth of the predator $s$ are located in the
hatched green area, then system (2) supports the stabilisation of the predator
and the prey populations. If the predation rate is located in the blue area,
then there are conditions for which the system supports the oscillation of
both populations, i.e., there is a stable limit cycle surrounding a positive
equilibrium point in the first quadrant. Moreover, there are conditions in the
$(q,s)$ parameter space (near the Hopf curve) for which populations can
coexist and also oscillate simultaneously, i.e., there are two limit cycles
surrounding a positive equilibrium point in the first quadrant. However, for
the sake of the presentation, this region is not included in the bifurcation
diagram. Instead, we refer to [saez]. Note that the extinction of (one of) the
species is not possible in this model.

###  2.2 Allee effect (strong and weak)

The May–Holling–Tanner model with a strong Allee effect ($m>0$) was studied in
[arancibia3]. The authors proved that system (4) with $m>0$ has three
equilibrium points in the $x$-axis and at most two positive equilibrium
points, see the left panel of Figure 6. The authors also proved that one of
the positive equilibrium points is always a saddle. While, the other can be
stable, unstable or stable surrounded by an unstable limit cycle. For $m<0$
there are two equilibrium points on the $x$-axis and at most three positive
equilibrium points, see the right panel of Figure 6. The stability of the
equilibrium points of system (4) considering $m<0$ is studied partially in
Appendix A.

First, we recall that as we used a diffeomorphism preserving the orientation
of time the nondimensionalised version (9) is topologically equivalent to the
original version of the model (4). Therefore, the results can be transferred
to the modified May–Holling–Tanner model. Moreover, we can extend the analysis
by using the system parameters $(r,K,a,n,c,m)=(4,150,6,0.025,0.01,\pm 15)$
(see Table 2). The bifurcation diagram of the May–Holling–Tanner model with
weak Allee effect (4) ($m<0$) and two typical phase portraits are shown in the
third row of Figure 3. System (4) with $m<0$ is not discussed in detail in
this manuscript, but the bifurcation diagram can be obtained by using the
techniques presented in [arancibia3]. If the parameters $(q,s)$ are located in
the hatched green region, then system (4) has one positive equilibrium point
which is stable, and thus both the predator and the prey population can
coexist. If the parameters $(q,s)$ are located in the blue region, then the
equilibrium point is unstable surrounded by a stable limit cycle and thus both
the predator and the prey populations oscillate.

However, if the parameters $(q,s)$ are located in the grey region, then system
(4) has three equilibrium points. These equilibrium points can be a saddle, a
stable and/or an unstable node. Therefore, system (4) supports the coexistence
and/or oscillation of the predator and prey populations. Note that the
dynamics of system (4) with $m<0$, outside the grey region, are qualitatively
similar to the dynamics of system (2) as shown in row one in Figure 3 since
there is a large region in the $(q,s)$ parameter space in which there is
always one positive equilibrium point, i.e. the hatched green region.

Additionally, the Allee effect, both strong and weak, decreases the range of
the value of the predation rate for which both populations can coexist, i.e.
$\tilde{q}_{2}<q_{1}$ (see Figure 3). Moreover, as in the original
May–Holling–Tanner model, the predator and prey populations never become
extinct in system (4) and thus there are always conditions in the system
parameters where both populations coexist (see hatched green area in Figure 3)
and/or oscillate (see blue and grey areas in Figure 3).

The bifurcation diagram of the modified May–Holling–Tanner model with strong
Allee effect (4) ($m>0$) studied in [arancibia3] and two typical phase
portraits are shown in the fourth row of Figure 3. The Hopf curve for the
model shows that if $(q,s)$ are located in the green area, system (4) supports
the stabilisation of both populations and also the extinction of both
populations. However, if the predation rate is located in the red hatched area
in Figure 3 ($q>\tilde{q}_{1}$), then system (4) does not have positive
equilibrium points in the first quadrant and thus the system only supports the
extinction of both the predator and the prey populations. In contrast, if the
predation rate and the intrinsic growth of the predator $(q,s)$ are located in
the solid red area, then system (4) has two positive equilibrium points which
are a saddle and an unstable node, and thus again we observe the extinction of
the predator and prey populations.

###  2.3 Alternative food

The model with alternative food for predators corresponds to system (6). This
model was studied in [arancibia7, arancibia2, arancibia5] where the authors
presented the stability of the equilibrium points and conditions for periodic
solutions. The bifurcation diagram of the modified May–Holling–Tanner model
(6) and two typical phase portraits are shown in the second row of Figure 3,
see also [arancibia9, arancibia5]. The Hopf curve for the modified
May–Holling–Tanner model shows that if the predation rate $q$ and the
intrinsic growth of the predator $s$ are located in the green area, then
system (6) supports the stabilisation of both populations and also the
extinction of only the prey population. If the predation rate is located in
the hatched brown area in Figure 3 ($q>q_{2}$), then system (6) does not have
a positive equilibrium point in the first quadrant and thus the extinction of
the prey population is observed, while the predator population stabilises at
$(0,c)$ (the equilibrium point $(0,c)$ is related to the alternative food
source for the predator).

Moreover, when the parameters $(q,s)$ are located in the solid brown area,
then system (6) has two positive equilibrium points in the first quadrant,
namely a saddle and an unstable node, and the extinction of the prey
population is again observed for all initial conditions. When the parameters
$(q,s)$ are located in the hatched brown area, then system (6) does not have
equilibrium points in the first quadrant and the extinction of the prey
population is observed for all initial conditions. Additionally, in
[arancibia9, arancibia5] the authors have shown that there is a region, near
to the Hopf bifurcation, in which both populations oscillate and coexist at
the same time, i.e. there are two limit cycles surrounding a positive
equilibrium point in the first quadrant. For the sake of the presentation, we
did not include this region in the bifurcation diagram.

###  2.4 Allee effect (strong and weak) and alternative food

The May–Holling–Tanner model with Allee effect on the prey and alternative
food for predators is given by system (7). This model was studied partially in
[arancibia] and in Appendix B. The bifurcation diagram of the modified
May–Holling–Tanner model with Allee effect on the prey and alternative food
for the predator (7) and two typical phase portraits are shown in the last row
of Figure 3. System (2) with Allee effect is not discussed in detail in this
manuscript, but the bifurcation diagram can also be obtained by using the
techniques presented in [arancibia3]. The Hopf curve for the system (7) with
Allee effect and alternative food shows that system (7) supports the
coexistence of both populations, the extinction of the prey population only
(green area) and the extinction of the prey population (solid and hatched
brown regions). That is, the behaviour is similar to the modified
May–Holling–Tanner model (6), see the second row of Figure 3.

In Figure 3, we show the bifurcation diagrams (first columns) in which
different regions represent different behaviours. In the solid green region,
sets of initial conditions exist that lead to the coexistence of both species,
while there are also open sets of initial conditions that lead to the
extinction of both species. In the hatched green regions, all initial
conditions lead to the coexistence of both populations. In the solid blue
regions, all initial conditions lead to the oscillation of both populations.
In the solid and hatched red regions, all initial conditions lead to the
extinction of both species, while in the solid and hatched brown regions, all
initial conditions lead to the extinction of the prey only. In the right
column, the orange region (dark and light) represents the basin of attraction
of the stabilisation of both populations, the light blue (light grey) region
represents the basin of attraction of the extinction of both (only the prey)
populations and the yellow region represent the oscillation of both
populations.

Figure 3: The phase plane and bifurcation diagrams of modified
May–Holling–Tanner models ( (4), (6), (7) and (2)) for the system parameters
$(r,K,a,n,c,m)=(4,150,6,0.025,0.01,\pm 15)$ (see Table 2) fixed and created
with the numerical bifurcation package MATCONT [matcont2].

Figure 4 shows the time series of the predator and prey populations in the
original May–Holling–Tanner model (7), the May–Holling–Tanner model with
alternative food for the predator (6) and the May–Holling–Tanner model with
Allee effect on the prey (4). The system parameters are as in Table 2 and
$(q,s)=(700,1.25)$. Note that these values lie inside the solid or hatched
green regions in Figure 3. Moreover, different predator and prey initial
densities are taken into account to highlight the different types of
behaviours. If the parameters $(q,s)$ are located in the blue region of Figure
3, then the original model supports oscillation of the solutions and it does
not depend on the initial conditions, see the first column of Figure 4. That
is, every initial density evolves to the coexistence of the species. If the
parameters $(q,s)$ are located in the green region of Figure 3, then the
May–Holling–Tanner model with strong Allee effect ($m>0$) supports both
coexistence and extinction of both species. That is, there exists an initial
density which evolves to the coexistence of the species, while there are also
initial conditions which evolve to the extinction of both species (see second
column of Figure 4).

If the parameters $(q,s)$ are located in the grey region of Figure 3, then the
May–Holling–Tanner model with weak Allee effect ($m<0$) supports the
stabilisation of both populations to two different equilibrium points. That
is, there exists an initial density which evolves to a coexistence point,
while there are also initial conditions which evolve to a lower coexistence
point (see third column of Figure 1). Finally, the modified May–Holling–Tanner
model with an alternative food source for the predator supports, depending on
the initial conditions, both oscillation and the extinction of the prey
species only. That is, there exists as initial density which evolves to a
limit cycle which represents the oscillation of the species, while there are
also initial conditions which evolve to the extinction of the prey and the
stabilisation of the predator population (see fourth column of Figure 1).

Figure 4: The characteristics of predator-prey populations in a log scale in
the original May–Holling–Tanner model (2), the May–Holling–Tanner model with
alternative food (6), the May–Holling–Tanner model with strong Allee effect
(4) ($m>0$) and the May–Holling–Tanner model with weak Allee effect (4)
($m<0$) over time ($t$). In all the cases, the dynamics are obtained through
numerical integration considering different initial densities and the system
parameters are as in Table (2) and $(q,s)=(700,1.25)$.

##  3 Conclusion

In this manuscript, the May–Holling–Tanner predator-prey model with functional
response Holling type II (2) was reviewed. We also considered that the prey
population is affected by strong and weak Allee effect (3) and the predator
population has an alternative food source (5). Using a diffeomorphism we
extend the analysis of a topologically equivalent system (4), (6), (7) and
(2).

We have also extended the analysis by using the system parameters presented in
[hanski, turchin], see Table 2. In addition, we have shown that the predation
rate per capita (parameter $q$) impacts the number of equilibrium points in
the first quadrant while the intrinsic growth rate of the predator (parameter
$s$) changes the behaviour of the positive(s) equilibrium point(s), see Figure
3. Therefore, it is clear that the self-regulation depends on the values of
these two parameters. Consequently, we have shown that the original
May–Holling–Tanner model with a strong Allee effect ($m>0$), a weak Allee
effect ($m<0$) and with both modifications, i.e. Allee effects (strong and
weak) for the prey and alternative food for the predator, can mean extinction
and coexistence of prey and predator populations, depending on particular
conditions. This is because the stable manifold of a saddle equilibrium point
or an unstable limit cycle act as a separatrix between the basin of attraction
of the origin and a stable equilibrium point in the first quadrant. However,
the original May–Holling–Tanner model with a weak Allee effect ($m<0$) and
with alternative food for the predator are present with the coexistence or
oscillation of the population. This is due to the existence of at least one
positive equilibrium point which can be either stable, unstable surrounded by
a stable limit cycle or stable surrounded by two limit cycles.

In Figure 5 we observe that the introduction of a density dependent phenomenon
in the prey population and/or an alternative food source for the predator
population have an impact on the predation rate (parameter $q$). The predation
rate at which the Hopf curve of the Holling–Tanner model with Allee effect (4)
is reduced, see the top right panel of Figure 5. In particular, the Hopf curve
of the model with strong Allee effect shows that system (4) can have up to two
positive equilibrium points which can collapse at $q=\tilde{q}_{1}$
(respectively for $m<0$ at $q=\tilde{q}_{2}$). Therefore, for
$q<\tilde{q}_{1}$ system (4) presents two positive equilibrium points and thus
the coexistence of both populations is observed [arancibia3].

Figure 5: In the top left panel we present the Hopf bifurcation curve of the
original May–Holling–Tanner model (2) (red curve). In the top right panel, we
present the Hopf bifurcation curve of the May–Holling–Tanner model with Allee
effect (4) (blue curve). While, in the bottom left panel we present the Hopf
bifurcation curve of the May–Holling–Tanner model with alternative food for
the predator (6) (grey curve) and in the bottom right panel the Hopf
bifurcation curve of the May–Holling–Tanner model with both modification (7)
(orange curve) is presented. In all panels we used the system parameters
$(r,K,a,n,c,m)=(4,150,6,0.025,0.01,\pm 15)$ (see Table 2) fixed and created
with the numerical bifurcation package MATCONT [matcont2].

In contrast, the Hopf curve of the model with week Allee effect ($m<0$) shows
that if $q>\tilde{q}_{2}$ then system (4) with $m<0$ has always one positive
equilibrium point which is stable or unstable surrounded by a stable limit
cycle and thus the coexistence or oscillation of both population is observed.
Note that the dynamic of system (4) with $m<0$ is similar with the dynamics of
system (2) since in both systems there is always one positive equilibrium
point. Consequently, there is a coexistence and/or oscillation of predator and
prey population.

We also consider the Hopf curve of the Holling–Tanner model with alternative
food for predators (6) and the system parameters
$(r,K,a,n,c,m)=(4,150,6,0.025,0.01)$ (see Table 2) fixed, see the bottom left
panel of Figure 5. In particular, the Hopf curve for the original
May–Holling–Tanner model, considering an alternative food for predators, shows
that the positive equilibrium points collapse at $q=q_{2}$. Therefore, for
$q<q_{2}$ system (6) present also two positive equilibrium points. One of them
is always a saddle point and the other can be stable or unstable surrounded by
stable limit cycle and thus the coexistence or oscillation of both population
is also observed [arancibia5, arancibia9]. Note that if $q>q_{2}$ the
equilibrium point related to the alternative food is globally stable.
Therefore, system (6) supports the stabilisation of only the predator
population and the extinction of the prey population.

Next, we consider the Hopf curve of the May–Holling–Tanner model with both
modification (7), i.e. a strong Allee effect on the prey and alternative food
for the predator, and the system parameters
$(r,K,a,n,c,m)=(4,150,6,0.025,0.01,\pm 15)$ (see Table 2) fixed, see the
bottom right panel of Figure 5. In particular, the modified system (7) can
have up to two positive equilibrium points which can collapse at
$q=\hat{q}_{1}$. Therefore, for $q<\hat{q}_{1}$ system (7) support the
coexistence of both population or the extinction of only the prey population.

In summary, the bifurcation diagrams of the models studied in [saez,
arancibia3, arancibia] are often qualitatively similar but their solutions
behave quantitatively differently. In other words, it is observed that there
are two groups of modes which support equivalent ecological behaviour due to
the addition of the modifications in the May–Holling–Tanner model. That is, an
alternative food source for the predator, a strong Allee effect or the
combination of both modifications support the coexistence and the extinction
of at least one of the species. In contrast, the original model and the model
with weak Allee effect ($m<0$) do not support the extinction of one of the
species.

## Acknowledgements

This project was fully supported by Universidad de las Américas (UDLA) Chile.
Especially, authors would like to give special thanks to Jaime Garcia for his
support in the edition of this manuscript.

## Conflict of interest

The authors declare that no conflict of interest.

## Appendices

##  Appendix A The modified May–Holling–Tanner model with weak Allee effect
(9) ($M<0$).

The May–Holling–Tanner model with a strong Allee effect ($m>0$) was studied in
[arancibia3] in which the authors simplified the system by introducing a
change of variable and time rescaling. This is given by the function
$\phi:\breve{\Omega}\times\mathbb{R}\rightarrow\Omega\times\mathbb{R}$, where
$\phi(u,v,\tau)=(N,P,t)=(Ku,nKv,\tau u(u+a/K)/rK)$. Moreover, by setting

$M:=\dfrac{m}{K}<1,~{}Q:=\dfrac{qn}{rK},~{}S:=\dfrac{s}{rK}~{}\mbox{and}~{}A=\dfrac{a}{K}$
(8)

into system (4) we obtain

$\displaystyle\dfrac{du}{d\tau}$ $\displaystyle=u^{2}((u+A)(1-u)(u-M)-Qv)\,,$
(9) $\displaystyle\dfrac{dv}{d\tau}$ $\displaystyle=Sv(u+A)(u-v)\,.$

Figure 6: The intersection of the predator nullcline (blue line) and the prey
nullcline (red curve) in the May–Holling–Tanner model with strong ($M>0$) and
weak ($M<0$) Allee effects by changing the parameter $Q$. The dark red circle
represents a saddle equilibrium point, the grey circles represent an
equilibrium point which can be stable or unstable, the black circle represents
an equilibrium point which is always stable, the orange circle represents an
equilibrium point which is always unstable and the green circle represents an
equilibrium point which is the collision of two equilibrium points.

To analyse the positive solution(s) of system (9) we consider three cases of
the intersection of the nullcline of system (9) and $M<0$. The $u$-nullclines
of system (9) are $u=0$ and $v=(u+A)(1-u)(u-M)/Q$, while the $v$-nullclines
are $v=0$ and $v=u$. Hence, the equilibrium points for system (9) are $(0,0)$,
$(1,0)$ and the point(s) $(u^{*},v^{*})$ with $v^{*}=u^{*}$ and where $u^{*}$
is determined by the solution(s) of

$\displaystyle
u^{3}-(M+1-A)u^{2}-(A(M+1)-Q-M)u+AM=u^{3}-H(A,M)u^{2}-L(A,M,Q)u+AM=0,$ (10)

where $H(A,M)=M+1-A$ and $L(A,M,Q)=A(M+1)-Q-M$. System (9) always has one
positive equilibrium point, which we denote by $P_{1}=(u_{1},u_{1})$ where
$0<u_{1}<1$. Factoring out $(u-u_{1})$ from the equation resultant (10) of the
intersection of the nullcline of system (9) [arancibia3], leaves as with
second order polynomial equation

$u^{2}-(1-A+M-u_{1})u-\dfrac{AM}{u_{1}}=0$ (11)

Therefore, if $1-A+M-u_{1}>0$ then the roots of (11) are given by

$u_{2,3}=\dfrac{1}{2}\left(1-A+M-u_{1}\pm\sqrt{\Delta}\right),~{}\text{with}~{}\Delta=(1-A+M-u_{1})^{2}+\dfrac{4AM}{u_{1}},$
(12)

such that $0<u_{2}<u_{3}<1$.

###  A.1 Stability of positive equilibrium points:

To determinate the stability of the positive equilibrium points $P_{1}$,
$P_{2}$, and $P_{3}$ of system (9) with $M<0$ we compute the Jacobian matrix
(13) at these equilibrium points

$J(u_{i},u_{i})=\begin{pmatrix}u_{i}^{2}((1-u_{i})(u_{i}-M)+(u_{i}+A)(1-u_{i})-(u_{i}+A)(u_{i}-M))&-Qu_{i}^{2}\\\
Su_{i}(u_{i}+A)&-Su_{i}(u_{i}+A)\end{pmatrix},$ (13)

with $i=1,2,3$. Thus, the trace and the determinant of the Jacobian matrix
(13) are

$\displaystyle\det{(P_{i})}$ $\displaystyle=Su_{i}^{2}(u_{i}+A)h(u_{i})$ (14)
$\displaystyle{\rm{tr}}{(P_{i})}$
$\displaystyle=u_{i}(u_{i}+A)\left(f(u_{i})-S\right),$

with

$\displaystyle h(u_{i})$
$\displaystyle=u_{i}^{2}(2u_{i}-(1-A+M))-AM~{}\text{and}$ (15) $\displaystyle
f(u_{i})$
$\displaystyle=\dfrac{u_{i}}{u_{i}+A}\left((1-u_{i})(u_{i}-M)+(u_{i}+A)(1-u_{i})-(u_{i}+A)(u_{i}-M)\right).$

Then, we consider the stability of these positive equilibrium points $P_{1}$,
$P_{2}$ and $P_{3}$ of system (9) with $M<0$ in the interior of
$\Phi=\\{\left(u,v\right),\ 0<u\leq 1,\ v\geq 0\\}$.

######  Lemma A.1.

Let the system parameters of (9) be such that $1-A+M\leq 0$ or $1-A+M>0$ and
$A(M+1)-Q-M\geq 0$. Then system (9) has only one positive equilibrium point
$(u_{1},u_{1})$ which can be

  1. (i)

a repeller if $S<f(u_{1})$,

  2. (ii)

an attractor if $S>f(u_{1})$,

with $f(u_{*})$ defined in (15).

###### Proof.

Evaluating $u^{2}(2u-(1-A+M))-AM$ at $u=u_{1}$ gives:

$u^{2}(2u-(1-A+M))-AM=u_{1}^{2}(2u_{1}-(1-A+M))-AM.$

Then,

  1. (i)

if $1-A+M<0$ then $u_{1}^{2}(2u_{1}-(1-A+M))-AM>0$. Hence $\det(J(P_{1}))>0$
since $M<0$,

  2. (ii)

if $1-A+M=0$ then $2u_{1}^{3}-AM>0$. Hence $\det(J(P_{1}))>0$,

  3. (iii)

if $1-A+M>0$ then
$u_{1}^{2}(2u_{1}-(1-A+M))-AM=u_{1}^{3}+(A(1+M)-Q-M)u_{1}-2AM>0$. Hence
$\det(J(P_{1}))>0$ since $A(M+1)-Q-M\geq 0$.

Therefore, the sign of the trace (14), and thus the behaviour of
$(u_{1},u_{1})$ depends on the parity of $f(u_{1})-S$. ∎

######  Lemma A.2.

Let the system parameters of (9) be such that $1-A+M>0$, $A(M+1)-Q-M<0$ and
$\Delta>0$ (12). Then the equilibrium point $P_{1}$ is

  1. (i)

a saddle point if $h(u_{1})<0$

  2. (ii)

a repeller if $h(u_{1})>0$ and $S<f(u_{1})$,

  3. (iii)

an attractor if $h(u_{1})>0$ and $S>f(u_{1})$.

with $f(u_{*})$ and $h(u_{*})$ defined in (15).

###### Proof.

Evaluating $u^{2}(2u-(1-A+M))-AM$ at $u=u_{1}$ gives:

$u^{2}(2u-(1-A+M))-AM=u_{1}^{2}(2u_{1}-(1-A+M))-AM.$

Then, the sign of the determinant and the trace (14), and thus the behaviour
of $P_{1}$ depends on the parity of $h(u_{1})$ and $f(u_{1})-S$. ∎

######  Lemma A.3.

Let the system parameters of (9) be such that $1-A+M>0$, $A(M+1)-Q-M<0$ and
$\Delta>0$ (12). Then the equilibrium point $P_{2}$ is

  1. (i)

a saddle point if $u_{2}>u_{1}$

  2. (ii)

a repeller if $u_{2}<u_{1}$ and $S<f(u_{2})$,

  3. (iii)

an attractor if $u_{2}<u_{1}$ and $S>f(u_{2})$,

with $f(u_{*})$ defined in (15).

###### Proof.

Evaluating $u^{2}(2u-(1-A+M))-AM$ at $u=u_{2}=(1-A+M-u_{1}-\sqrt{\Delta})/2$
gives:

$\displaystyle u^{2}(2u-(1-A+M))-AM=$
$\displaystyle\dfrac{\sqrt{\Delta}}{4}(-1-A-M+3u_{1}+\sqrt{\Delta})$
$\displaystyle(1-A+M-u_{1}-\sqrt{\Delta})=u_{2}\sqrt{\Delta}(u_{1}-u_{2}).$

Then, the sign of the determinant depends on the parity of $u_{1}-u_{2}$.
Then, the trace (14), and thus the behaviour of $P_{2}$ depends on the parity
of $f(u_{2})-S$. ∎

######  Lemma A.4.

Let the system parameters of (9) be such that $1-A+M>0$, $A(M+1)-Q-M<0$ and
$\Delta>0$ (12). Then the equilibrium point $P_{3}$ is

  1. (i)

a saddle point if $u_{3}<u_{1}$

  2. (ii)

a repeller if $u_{3}>u_{1}$ and $S<f(u_{3})$,

  3. (iii)

an attractor if $u_{3}>u_{1}$ and $S>f(u_{3})$,

with $f(u_{*})$ defined in (15).

###### Proof.

Evaluating $u^{2}(2u-(1-A+M))-AM$ at $u=u_{3}=(1-A+M-u_{1}+\sqrt{\Delta})/2$
gives:

$\displaystyle u^{2}(2u-(1-A+M))-AM=$
$\displaystyle\dfrac{\sqrt{\Delta}}{4}(-1+A-M+3u_{1}-\sqrt{\Delta})$
$\displaystyle(-1+A-M+u_{1}-\sqrt{\Delta})=u_{3}\sqrt{\Delta}(u_{3}-u_{1}).$

Then, the sign of the determinant depends on the parity of $u_{3}-u_{1}$.
Then, the trace (14), and thus the behaviour of $P_{3}$ depends on the parity
of $f(u_{3})-S$. ∎

##  Appendix B The modified May–Holling–Tanner model with alternative food for
the predator and Allee effect.

The May–Holling–Tanner model with Allee effect on the prey and alternative
food for predators is given by system (7). The system can be simplified by
introducing a change of variable and time rescaling presented in [arancibia].
It is given by the function
$\zeta:\breve{\Psi}\times\mathbb{R}\rightarrow\Psi\times\mathbb{R}$, where
$\zeta(u,v,\tau)=(N,P,t)=(Ku,nKv,\tau(u+a/K)(u+c/(nK))/rK)$ [arancibia]. By
setting the parameters $A$, $C$, $Q$, $S$ and $M$ defined in (8), into system
(7) we obtain

$\displaystyle\dfrac{du}{d\tau}$ $\displaystyle=u(u+C)((u+A)(1-u)(u-M)-Qv)\,,$
(16) $\displaystyle\dfrac{dv}{d\tau}$ $\displaystyle=Sv(u+A)(u-v+C)\,,$

Figure 7: The intersection of the predator nullcline (blue line) and the prey
nullcline (red curve) in the May–Holling–Tanner model with an alternative food
for predators and strong ($M>0$) and weak ($M<0$) Allee effect by changing the
parameter $Q$. The purple circle represents a saddle equilibrium point, the
grey circle represents an equilibrium point which can be stable or unstable,
the black circle represents an equilibrium point which is always stable and
the orange circle represents an equilibrium point which is always unstable.

To analyse the positive solution(s) of system (16) we consider three cases of
the intersection of the nullcline of system (16). The $u$-nullclines of system
(16) are $u=0$ and $v=(u+A)(1-u)(u-M)/Q$, while the $v$-nullclines are $v=0$
and $v=u+C$. Hence, the equilibrium points for system (16) are $(0,0)$,
$(1,0)$, $(M,0)$ and the point(s) $(u^{*},v^{*})$ with $v^{*}=u^{*}+C$ and
where $u^{*}$ is determined by the solution(s) of

$u^{3}-(M+1-A)u^{2}-(A(M+1)-Q-M)u+AM+CQ=0.$ (17)

Define the functions $g(u)=(u+A)(1-u)(u-M)$ and $h(u)=Q(u+C)$ and observe that
$\lim\limits_{u\rightarrow\pm\infty}g(u)=\mp\infty$ and $g(0)=-AM<0$. So, (17)
will always have a single negative real root, which we denote by $u=-G$ where
$G>0$. Factoring out $(u+G)$ from (17) leaves us with second order polynomial

$u^{2}-(G+M+1-A)u+(M+Q-A(M+1)+G(G-A+M+1))=0,$ (18)

and thus we get $G(M+Q-A(M+1)+G(G-A+M+1))-AM-CQ=0$. Note that since $Q>0$,
then we obtain that $A<G<C$ or else, $A>G>C$. In addition, we get that
$Q=(G+1)(G+M)(G-A)/(C-G)$. The solution of (18) are:

$\displaystyle u_{1,2}$
$\displaystyle=\dfrac{1}{2}\left(1-A+G+M\pm\sqrt{\Delta}\right)\,,\quad\text{with}$
(19) $\displaystyle\Delta$
$\displaystyle=(G-A+M+1)^{2}-4(M+Q-A(M+1)+G(G-A+M+1))\,,$

such that $M<u_{1}\leq E\leq u_{2}<1$, where $E=(1-A+M+G)/2$. In particular,

  1. 1.

if $\Delta<0$ (19), then (16) has no equilibrium points in the first quadrant;

  2. 2.

if $\Delta>0$ (19), then (16) has two equilibrium points
$P_{1,2}=(u_{1,2},u_{1,2}+C)$ in the first quadrant; and

  3. 3.

if $\Delta=0$ (19), then (16) has one equilibrium point $(E,E+C)$ of order two
in the first quadrant,

Additionally, in Figure 7 we observe that by changing the parameter $Q$ and if
$M<0$ (weak Allee effect), then system (16) also has no equilibrium points,
two equilibrium points or one equilibrium point of order two in the first
quadrant.

###  B.1 Stability of positive equilibrium points

To determine the nature of the equilibrium points we compute the Jacobian
matrix $J(u,v)$ of (16)

$J(u,v)=\begin{pmatrix}u(u+C)g^{\prime}(u)+(2u+C)(g(u)-Qv)&-Qu(u+C)\\\
Sv(A+C+2u-v)&S(C+u-2v)(u+A)\end{pmatrix},$ (20)

with $g(u)=(u+A)(1-u)(u-M)$ and
$g^{\prime}(u)=(1-u)(u-M)+(u+A)(1-u)-(u+A)(u-M)$.

######  Lemma B.1.

The equilibrium point $(0,0)$ is a saddle point, $(1,0)$ is a saddle point,
$(M,0)$ is an unstable point and $(0,C)$ is a stable point.

###### Proof.

The Jacobian matrix (20) evaluated at $(0,0)$ and $(1,0)$ gives

$J(0,0)=\begin{pmatrix}-ACM&0\\\ 0&ACS\end{pmatrix}~{}\text{and}$
$J(1,0)=\begin{pmatrix}-(1-M)(C+1)(A+1)&-Q(C+1)\\\
0&S(C+1)(A+1)\end{pmatrix}.$

Similarly, the Jacobian matrix (20) evaluated at $(M,0)$ gives

$J(M,0)=\begin{pmatrix}M(1-M)(C+M)(A+M)&-MQ(C+M)\\\
0&S(A+M)(C+M)\end{pmatrix}.$

Finally, the Jacobian matrix (20) evaluated at $(0,C)$ gives

$J(0,C)=\begin{pmatrix}-C(AM+QC)&0\\\ ACS&-ACS\end{pmatrix}.$

Since $M<1$, it follows that $(0,0)$ and $(1,0)$ are a saddle point, $(M,0)$
is an unstable point and $(0,C)$ is a stable point in system (16). ∎

Next, we consider the stability of the positive equilibrium points $P_{1}$ and
$P_{2}$ of system (16). The Jacobian matrix (20) of system (16) at these
equilibrium points becomes

$J(u,u+C)=\begin{pmatrix}u(u+C)g^{\prime}(u)&-Qu(u+C)\\\
S(u+C)(u+A)&-S(u+C)(u+A)\end{pmatrix},$ (21)

Here, the determinant and the trace of the Jacobian matrix (21) are given by

$\displaystyle\det(J(u,u+C))$ $\displaystyle=Su(u+A)u+C)^{2}(Q-g^{\prime}(u))$
(22) $\displaystyle{\rm{tr}}(J(u,u+C))$
$\displaystyle=(u+C)(u+A)\left(\dfrac{ug^{\prime}(u)}{u+A}-S\right)$

Therefore, the sign of the determinant the trace (22), and thus the stability
of the positive equilibrium points, depends on the sign of
$Q-g^{\prime}(u_{1,2})$ and $ug^{\prime}(u)/(u+A)-S$ respectively. This gives
the following results.

######  Lemma B.2.

Let the system parameters of (16) be such that $\Delta>0$ (22). Then, the
equilibrium point $P_{1}$ is a saddle point.

###### Proof.

Evaluating $Q-g^{\prime}(u)$ at $u_{1}$ gives:

$Q-g^{\prime}(u_{1})=-\dfrac{\sqrt{\Delta}}{2}\left(3G+M+1-A-\sqrt{\Delta}\right)$

Hence, $\det(J(P_{1}))<0$ and $P_{1}$ is thus a saddle point. ∎

######  Lemma B.3.

Let the system parameters be such that $\Delta>0$ (22). Then, the equilibrium
point $P_{2}$ is:

  1. 1.

unstable if $S<\dfrac{D}{4\left(1+A+M+G+\sqrt{\Delta}\right)}$; and

  2. 2.

stable if $S>\dfrac{D}{4\left(1+A+M+G+\sqrt{\Delta}\right)}$,

with
$D=\left(1-A+M+G+\sqrt{\Delta}\right)\left(1+A-M-G-\sqrt{\Delta}\right)\left(1-A-M+G+\sqrt{\Delta}\right)+2\left(1+A+M+G+\sqrt{\Delta}\right)\left(A-G-\sqrt{\Delta}\right)$.

###### Proof.

Evaluating $Q-g^{\prime}(u)$ at $u_{2}$ gives:

$Q-g^{\prime}(u_{1})=-\dfrac{\sqrt{\Delta}}{2}\left(3G+M+1-A+\sqrt{\Delta}\right)$

Hence, $\det(J(P_{2}))>0$. Evaluating $ug^{\prime}(u)/(u+A)-S$ at $u_{2}$
gives

$\displaystyle\dfrac{u_{2}g^{\prime}(u_{2})}{u_{2}+A}-S=$
$\displaystyle\dfrac{(1-A+M+G+\sqrt{\Delta})(1+A-M-G-\sqrt{\Delta})(1-A-M+G+\sqrt{\Delta})}{4(1+A+M+G+\sqrt{\Delta})}$
$\displaystyle+\dfrac{(A-G-\sqrt{\Delta})}{2}-S$

Therefore, the sign of the trace, and thus the behaviour of $P_{2}$ depends on
the sign of $u_{2}g^{\prime}(u_{2})/(u_{2}+A)-S$. ∎

Next we analyse the case when $\Delta=0$ (22) then the equilibrium points
$P_{1}$ and $P_{2}$ collapse such that $u_{1}=u_{2}=E=(1-A+M+G)/2$. Therefore,
system (16) has one equilibrium point of order two in the first quadrant given
by $(E,E+C)$.

######  Lemma B.4.

Let the system parameters be such that $\Delta=0$ (19). Then, the equilibrium
point $(E,E+C)$ is:

  1. 1.

a stable saddle-node if $S<\dfrac{H}{4(1+A+M+G)}$; and

  2. 2.

an unstable saddle-node if $S>\dfrac{H}{4(1+A+M+G)}$,

with $H=(1-A+M+G)(1+A-M-G)(1-A+G-M)+2(1+M+G+A)(A-G)$.

###### Proof.

Evaluating $Q-g^{\prime}(u)$ at $u=E$ gives

$Q-g^{\prime}(E)=0.$

Therefore, $\det(J(E,E))=0$. Next, evaluating $ug^{\prime}(u)/(u+A)-S$ at $E$
gives

$\displaystyle Eg^{\prime}(E)/(E+A)-S=$
$\displaystyle\dfrac{(1-A+M+G)(1+A-M-G)(1-A-M+G)}{4(1+A+M+G)}$
$\displaystyle+\dfrac{(A-G)}{2}-S.$

Therefore, the behaviour of the equilibrium point $(E,E+C)$ depends on the
parity of $Eg^{\prime}(E)/(E+A)-S$. ∎

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