Bistability and Hopf bifurcation of a tritrophic system with Holling functional responses

Abstract

In this paper we analyze the dynamics of a tritrophic food chain, with functional responses of Holling type III and II for the mesopredator and superpredator respectively and logistic growth rate for the prey. We show that there are parameter conditions for which the system has up to three equilibrium points. When three equilibrium points appear we show that each one of them exhibits a supercritical Hopf bifurcation. Moreover we can also show that there is a set of parameters for which the system exhibits a simultaneous Hopf bifurcation.

Hopf bifurcation; Limit cycle; Tritrophic model

1. Introduction

One of the main problems in ecology is trying to comprehend the different interaction mechanisms that allows the coexistence of species, one of them is predation. In the past decades, based on the Lotka- Volterra predator prey model, various mathematical models have appeared in the literature. These mathematical models have permitted us to analyze the growth of the species involved in the model (May, 1973, Holling, 1959, Hsu and Huang, 1995, Valenzuela et al., 2017).

In the literature we can find several articles studying tritrophic models, in them the authors analyzed the dynamics of the system in order to establish conditions under which the species can coexist. Furthermore they analyzed and determined conditions in order to obtain a stable limit cycle in tritrophic models. These are systems in which three differential equations are involved, as they analyze the behavior of three species, prey, predator and superpredator (Francoise and Llibre, 2011, Castellanos et al., 2018, Dawed et al., 2020) and the references therein.

In the works cited above, the main focus has been the effect the predator has in the growth of the prey. This effect is measured by a functional response, which can be of type Holling, Beddington-DeAngelis, Crowley-Martin, Hassell-Varley, etc. (Skalski and Gilliam, 2001).

As a result of the analysis of the tritrophic models, the authors presented conditions in the parameters of the analyzed system, under which there exists a stable limit cycle for a system with a functional response of type Lotka Volterra or Holling. This was proved by showing the existence of a Hopf or a Bautin bifurcation (Castellanos et al., 2018, Bentounsi et al., 2018, Blé et al., 2018, Wang and Yu, 2019, Dawed et al., 2020).

A tritrophic food chain is modeled by the following system of three differential equations:

$$\frac{dx}{dt}=h(x)-f(x)y,\frac{dy}{dt}=c_{1}yf(x)-g(y)z-c_{2}y,\frac{dz}{dt}=c_{3}g(y)z-d_{2}z,$$ (1)

where x represents the density of a prey that gets eaten by a predator of density y, and the species y feeds the top predator z (superpredator). The function $h(x)$ represents the growth rate of the prey population in absence of the predators, and the functions $f(x)$ and $g(y)$ are the functional responses for the mesopredator and the superpredator, respectively (Castellanos et al., 2018). The parameters $c_1,c_2,c_3$ and $d_2$ are positive and for ecological considerations we focus in finding stable solutions in the positive octant

$$\mathrm{\Omega }=\{x>0,y>0,z>0\}.$$

In this paper we consider functional responses $f(x)$ and $g(y)$ of Holling type III and II respectively. Explicitly

$$f(x)=\frac{a_1{x}^2}{b_1+x^2}\text{ and }g(y)=\frac{a_{2}y}{b_2+y},$$

where $a_1,b_1,a_2,b_2$ are positive constants. For the prey we consider a logistic growth, for this we take $h(x)=\rho x(1-\frac{x}{k})$.

Then system (1) becomes:

$$\frac{dx}{dt}=\rho x(1-\frac{x}{k})-\frac{a_1{x}^{2}y}{x^2+b_1},\frac{dy}{dt}=\frac{c_1{a}_{1}y{x}^2}{x^2+b_1}-\frac{a_{2}yz}{b_2+y}-c_{2}y,\frac{dz}{dt}=\frac{c_3{a}_{2}yz}{b_2+y}-d_{2}z.$$ (2)

In this work, to determine the coexistence of the three populations, we first establish conditions in the parameters for which there is an equilibrium point. Then we find conditions for which a supercritical Hopf bifurcation occurs and consequently stable limit cycles are calculated. We also provide parameter values where the conditions given by the results are satisfied.

2. Equilibrium points in the positive octant

In order to describe the coexistence equilibrium points of the system (2) we give the following Lemma.

Lemma 2.1

A point $p_0=(x_0,y_0,z_0)\in \mathrm{\Omega }$ is an equilibrium of system (2) if and only if the following conditions hold

$$\begin{gathered} y_0=\frac{b_2{d}_2}{a_2{c}_3-d_2} \\ {z}_0=\frac{b_2{c}_2{c}_3}{d_2-a_2{c}_3}+\frac{c_1{c}_3\rho x_0(k-x_0)}{d_{2}k} \\ {a}_1=\frac{\rho (b_1+x_0^2)(k-x_0)(a_2{c}_3-d_2)}{b_2{d}_{2}k{x}_0}. \end{gathered}$$ (3)

Proof

An equilibrium point for the system (2) is a solution of the system of equations given by (2) where the derivatives vanish. Multiplying each equation by $(b_1+x^2)(b_2+y)$, we obtain

$$\begin{array}{ccc}\hfill 0& \hfill =\hfill & \frac{x(b_2+y)(\rho (b_1+x^2)(k-x)-a_{1}kxy)}{k},\hfill \\ \hfill 0& \hfill =\hfill & -y(b_1{b}_2{c}_2-a_1{b}_2{c}_1{x}^2+b_2{c}_2{x}^2+b_1{c}_{2}y-a_1{c}_1{x}^{2}y+c_2{x}^{2}y\hfill \\ \hfill & \hfill \hfill & +a_2{b}_{1}z+a_2{x}^{2}z),\hfill \\ \hfill 0& \hfill =\hfill & -(b_1+x^2)(-a_2{c}_{3}y+d_2(b_2+y))z.\hfill \end{array}$$ (4)

Since the parameters $b_1,b_2$ are positive, all solutions $(x_0,y_0,z_0)\in \mathrm{\Omega }$ of the system (4) are coexistence equilibrium points of the system (2) if they are solutions of:

$$\begin{array}{ccc}\hfill 0& \hfill =\hfill & (k-x)(b_1+x^2)\rho -a_{1}kxy,\hfill \\ \hfill 0& \hfill =\hfill & b_1{b}_2{c}_2-a_1{b}_2{c}_1{x}^2+b_2{c}_2{x}^2+b_1{c}_{2}y-a_1{c}_1{x}^{2}y+c_2{x}^{2}y\hfill \\ \hfill & \hfill \hfill & +a_2{b}_{1}z+a_2{x}^{2}z,\hfill \\ \hfill 0& \hfill =\hfill & a_2{c}_{3}y-d_{2}y-b_2{d}_2.\hfill \end{array}$$ (5)

Notice that the first equation is a cubic polynomial in terms of x, with leading coefficient $-\rho <0$ and constant coefficient $k{b}_1\rho >0$. Thus always have a positive root $x_0$. Since the first and third equation of the system (5) are linear in terms of $a_1$ and y, respectively. They have the unique solutions:

$$a_1=\frac{\rho (b_1+x_0^2)(k-x_0)(a_2{c}_3-d_2)}{b_2{d}_{2}k{x}_0}.$$

$$y_0=\frac{b_2{d}_2}{a_2{c}_3-d_2}.$$

Substituting these values in the second equation we obtain:

$$0=a_2(b_1+x_0^2)(\frac{b_2{c}_2{c}_3}{a_2{c}_3-d_2}+\frac{c_1{c}_3\rho x_0(x_0-k)}{d_{2}k}+z).$$

Since $a_2$ and $b_1$ are positive solving in terms of z the solution is:

$$z_0=\frac{b_2{c}_2{c}_3}{d_2-a_2{c}_3}+\frac{c_1{c}_3\rho x_0(k-x_0)}{d_{2}k}.$$

 □

Corollary 2.2

If

$$a_2=\frac{d_2+k_1}{c_3},b_2=\frac{c_1{k}_1(k-x_0)x_0\rho }{2c_2{d}_{2}k},\mathit{\text{and}}k=x_0+k_2$$ (6)

where $k_1,k_2$ are positive real numbers, then $y_0,z_0$ given by Lemma 2.1 are positive and the coexistence equilibrium point $p_0$ can be written as:

$$p_0=(x_0,\frac{c_1{k}_2{x}_0\rho }{2c_{2}k},\frac{c_1{c}_3{k}_2{x}_0\rho }{2d_{2}k}).$$

Proposition 2.3

If the parameters satisfy (3), (6) and:

$$b_1=\frac{x_0(-1+{(1+x_0)}^2)}{4(1+2x_0)},\mathit{\text{and}}{k}_2=1+x_0.$$

Then the system (2), has three coexistence equilibrium points in Ω, given by:

$$p_0=(x_0,\frac{c_1{x}_0(1+x_0)\rho }{2(c_2+2c_2{x}_0)},\frac{c_1{c}_3{x}_0(1+x_0)\rho }{2(d_2+2d_2{x}_0)}),p_1=(\frac{x_0}{2},\frac{c_1{x}_0(1+x_0)\rho }{2(c_2+2c_2{x}_0)},\frac{c_1{c}_3{x}_0^2\rho }{4d_2+8d_2{x}_0}),p_2=(\frac{2+x_0}{2},\frac{c_1{x}_0(1+x_0)\rho }{2(c_2+2c_2{x}_0)},\frac{c_1{c}_3{x}_0(4+x_0)\rho }{4(d_2+2d_2{x}_0)}).$$ (7)

Proof

By Lemma 2.1 the possible equilibrium points $p=(x,y,z)$ of the system (2) are of the form

$$p=(x,\frac{b_2{d}_2}{a_2{c}_3-d_2},\frac{b_2{c}_2{c}_3}{d_2-a_2{c}_3}+\frac{c_1{c}_3\rho x(k-x)}{d_{2}k}).$$

Thus we only need to find the roots of first equation of (5) which is a cubic polynomial in x. As $x_0$ is a root of this polynomial, Lemma 2.1 implies that

$$a_1=\frac{\rho (b_1+x_0^2)(k-x_0)(a_2{c}_3-d_2)}{b_2{d}_{2}k{x}_0}.$$

Substituting $a_1$ and $y=y_0$ as in (3) the cubic polynomial becomes:

$$P(x):=-b_{1}k\rho +\frac{\rho x(b_{1}k+k{k}_2^2-x_0^3)}{x_0}-k\rho x^2+\rho x^3.$$

Since $P(x_0)=0$, we can write:

$$P(x)=\frac{\rho (x-x_0)(b_{1}k-kx{x}_0+x^2{x}_0+x{x}_0^2)}{x_0}.$$ (8)

For the given values of $b_1$ and $k_2$ we can reduce this expression to:

$$P(x)=\frac{1}{4}\rho (x-x_0)(2x-x_0-2)(2x-x_0),$$

which has solutions $\{x_0,\frac{x_0+2}{2},\frac{x_0}{2}\}$. Thus the result follows. □

Notice the following. In (8) we obtain a cubic polynomial whose roots give us the three equilibrium points that we are going to analyze in the next section. However, we are going to focus on the possibility of roots of this polynomial. We already know that $P(x)$ has a root $x_0$. Given the factorization in (8) we need to focus in the second degree polynomial:

$$(b_1{k}_2+b_1{x}_0-k_2{x}_{0}x+x_0{x}^2).$$

The discriminant of this polynomial is:

$$x_0(k_2^2{x}_0-4b_1(k_2+x_0)).$$

Clearly if $b_1=\frac{k_2^2{x}_0}{4(k_2+x_0)}$ the polynomial will have a root with multiplicity two. In fact the root is $\frac{k_2}{2}$, in this case the equilibrium point arising from the root $\frac{k_2}{2}$ is not hyperbolic.

Also for any value of $b_1>\frac{k_2^2{x}_0}{4(k_2+x_0)}$ the cubic polynomial $P(x)$ have only one root. Choosing for example the value $b_1=\frac{k_2^2{x}_0}{2(k_2+x_0)}$, it is straightforward to obtain the characteristic polynomial of the linear approximation of the system at the only equilibrium point, which is

$$\begin{gathered} {\lambda }^3+{\lambda }^2(-\frac{(c_2{d}_2(k_2^2+2k_2{x}_0+2x_0^2)-\rho (d_2+k_1)(k_2^2-2k_2{x}_0+2x_0^2))}{(d_2+k_1)(k_2^2+2k_2{x}_0+2x_0^2)}) \\ +\lambda \left(\frac{c_2(d_2(k_1(k_2^3+3k_2^2{x}_0+4k_2{x}_0^2+2x_0^3)+\rho (3k_2^3+k_2^2{x}_0-2x_0^3))+4k_1{k}_2^3\rho )}{(d_2+k_1)(k_2+x_0)(k_2^2+2k_2{x}_0+2x_0^2)}\right) \\ +\frac{c_2{d}_2{k}_1\rho (k_2^2-2k_2{x}_0+2x_0^2)}{(d_2+k_1)(k_2^2+2k_2{x}_0+2x_0^2)} \end{gathered}$$

Using Mathematica and the Hurwitz criteria we have the following result.

Proposition 2.4

If $d_2<M_1=\frac{4k_1{k}_2^3}{2x_0^3-3k_2^3-k_2^2{x}_0}$ and

$$c_2<M_2=\frac{{\rho }^2(d_2+k_1)(k_2^2-2k_2{x}_0+2x_0^2)(k_2^3(3d_2+4k_1)+d_2{k}_2^2{x}_0-2d_2{x}_0^3)}{d_2(k_2^2+2k_2{x}_0+2x_0^2)(d_2{k}_1(k_2+x_0)(k_2^2+2k_2{x}_0+2x_0^2)+d_2\rho (3k_2^3+k_2^2{x}_0-2x_0^3)+4k_1{k}_2^3\rho )}$$

then $p_0$ is the only coexistence equilibrium point and it is locally asymptotically stable.

In order to illustrate that the parameter conditions in Proposition 2.4 are satisfied, we take

$$k_1=3,k_2=\frac{1}{2},\rho =x_0=c_1=c_3=1,$$

and we obtain $M_1=\frac{12}{11}$, choosing $d_2=1$, we have $M_2=\frac{10}{767}$. Finally, with $c_2=\frac{1}{128}$ the unique coexistence equilibrium point of the system is $p_0=(1,\frac{64}{3},\frac{1}{6})$ and it is locally asymptotically stable.

3. Hopf bifurcation at the equilibrium points of the system

In order to find periodic solutions we will prove that the system exhibits a Hopf bifurcation at each of the equilibrium points. To achieve this, we will calculate the first Lyapunov coefficient, furthermore we will find specific relations between the parameters to reduce the system to something more manageable. All the computations needed for the proofs were done using the software Mathematica.

3.1. The case for $p_0$

Choosing the values given by Corollary 2.2, the system becomes:

$$\dot{x}=x(-\frac{(2c_{2}x(b_1+x_0^2)y)}{(c_1(b_1+x^2)x_0^2)}+\rho -\frac{x\rho }{(k_2+x_0)}),\dot{y}=c_{2}y(-1+\frac{2x^2(b_1+x_0^2)}{(b_1+x^2)x_0^2}-\frac{2d_2(d_2+k_1)(k_2+x_0)z}{2c_2{c}_3{d}_2(k_2+x_0)y+c_1{c}_3{k}_1{k}_2{x}_0\rho }),\dot{z}=\frac{d_2{k}_{1}z(2c_2(k_2+x_0)y-c_1{k}_2{x}_0\rho )}{2c_2{d}_2(k_2+x_0)y+c_1{k}_1{k}_2{x}_0\rho }.$$

And

$$p_0=(x_0,\frac{c_1{k}_2{x}_0\rho }{2c_{2}k},\frac{c_1{c}_3{k}_2{x}_0\rho }{2d_{2}k})$$

is a coexistence equilibrium point of the system (2). The Jacobian matrix of the system (2) at the point $p_0$ is:

$$M=\left(\begin{array}{ccc}\hfill -\frac{((x_0^2(-k_2+x_0)+b1(k_2+x_0))\rho }{((k_2+x_0)(b_1+x_0^2))}\hfill & \hfill -\frac{2c_2}{c_1}\hfill & \hfill 0\hfill \\ \hfill \frac{2b_1{c}_1{k}_2\rho }{(k_2+x_0)(b_1+x_0^2)}\hfill & \hfill \frac{c_2{d}_2}{d_2+k_1}\hfill & \hfill -\frac{d_2}{c_3}\hfill \\ \hfill 0\hfill & \hfill \frac{c_2{c}_3{k}_1}{d_2+k_1}\hfill & \hfill 0\hfill \end{array}\right).$$

It follows from Lemmas 0.4 and 0.5 of the Appendix (Blé et al., 2021) that the corresponding expressions for α, ${\omega }^2$ and $\mathbf{D}(M)$ are

$$\alpha =\frac{c_2{d}_2}{d_2+k_1}-\frac{\rho (b_1(k_2+x_0)+x_0^2(x_0-k_2))}{(b_1+x_0^2)(k_2+x_0)},{\omega }^2=\frac{c_2(\frac{\rho (b_1(3d_2{k}_2-d_2{x}_0+4k_1{k}_2)+d_2{x}_0^2(k_2-x_0))}{(b_1+x_0^2)(k_2+x_0)}+d_2{k}_1)}{d_2+k_1},$$ (9)

$$\mathbf{D}(M)=\frac{c_2({\mathcal{H}}_1-c_2{\mathcal{H}}_2)}{{(b_1+x_0^2)}^2{(d_2+k_1)}^2{(k_2+x_0)}^2},$$ (10)

where

$${\mathcal{H}}_1={\rho }^2(d_2+k_1)(b_1(k_2+x_0)+x_0^2(x_0-k_2))(b_1(3d_2{k}_2-d_2{x}_0+4k_1{k}_2)+d_2{x}_0^2(k_2-x_0)),{\mathcal{H}}_2=d_2(b_1+x_0^2)(k_2+x_0)(\rho (b_1(3d_2{k}_2-d_2{x}_0+4k_1{k}_2)+d_2{x}_0^2(k_2-x_0))+d_2{k}_1(b_1+x_0^2)(k_2+x_0)).$$

Directly, we have the next result.

Lemma 3.1

Assuming that all the parameters involved in the system (2) are non zero, then $\mathbf{D}(M)=0$ if and only if the parameter $c_2=\frac{{\mathcal{H}}_1}{{\mathcal{H}}_2}$.

Remark 1

If $k_2=x_0$, then the parameters of the system (2) are positive and their respective values are:

In summary,

Proposition 3.2

If the hypothesis of Corollary 2.2 is satisfied, $k_2=x_0$ and

$$c_2=\frac{b_1^2{\rho }^2(d_2+k_1)(d_2+2k_1)}{d_2(b_1+x_0^2)(b_1\rho (d_2+2k_1)+d_2{k}_1(b_1+x_0^2))}$$

then the eigenvalues of the Jacobian matrix of the system (2) are

$$\alpha =-\frac{b_1{d}_2{k}_1\rho }{b_1\rho (d_2+2k_1)+d_2{k}_1(b_1+x_0^2)}<0,i\omega ,-i\omega ,$$

where $\omega =\frac{b_1\rho \sqrt{\frac{2k_1}{d_2}+1}}{b_1+x_0^2}>0$.

The system (2) with the assumptions of Proposition 3.2 becomes:

$$\dot{x}=x(\frac{x(-\frac{4c_{2}y(b_1+x_0^2)}{c_1(b_1+x^2)}-\rho x_0)}{2x_0^2}+\rho )\dot{y}=c_{2}y(\frac{2x^2(b_1+x_0^2)}{x_0^2(b_1+x^2)}-\frac{4d_{2}z(d_2+k_1)}{c_3(c_1{k}_1\rho x_0+4c_2{d}_{2}y)}-1)\dot{z}=\frac{d_2{k}_{1}z(4c_{2}y-c_1\rho x_0)}{c_1{k}_1\rho x_0+4c_2{d}_{2}y}$$ (11)

In order to compute the first Lyapunov coefficient, from now on in this section we fixed the parameters $b_1=\rho =d_2=k_1=c_1=x_0=c_3=1$. The next result is proved via a direct calculation.

Proposition 3.3

With the assumptions of the Proposition 3.2, the characteristic polynomial of the Jacobian matrix of system (11) is

$$p(\lambda (c_2))=\frac{1}{4}(2c_2-2){\lambda }^2-\frac{5c_2\lambda }{4}-\frac{c_2}{4}-{\lambda }^3.$$

The parameter of bifurcation is $c_2=\frac{3}{5}$. Furthermore the derivative of the real part of complex eigenvalues of the Jacobian matrix of system (11) evaluated at $c_2=\frac{3}{5}$ is $\frac{75}{316}$.

Proof

With the parameter values assigned below, the system (11) becomes

$$\dot{x}=\frac{1}{2}x(x(-\frac{8c_{2}y}{x^2+1}-1)+2)\dot{y}=c_{2}y(-\frac{8z}{4c_{2}y+1}+\frac{4x^2}{x^2+1}-1)\dot{z}=\frac{z(4c_{2}y-1)}{4c_{2}y+1},$$

the equilibrium point is $p_0=(1,\frac{1}{4c_2},\frac{1}{4})$ and the Jacobian matrix of the system is:

$$M(p_0)=\left(\begin{array}{ccc}\hfill -\frac{1}{2}\hfill & \hfill -2c_2\hfill & \hfill 0\hfill \\ \hfill \frac{1}{2}\hfill & \hfill \frac{c_2}{2}\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill \frac{c_2}{2}\hfill & \hfill 0\hfill \end{array}\right).$$

A straightforward calculation gives us that

$$p(\lambda )=\frac{1}{4}(2c_2-2){\lambda }^2-\frac{5c_2\lambda }{4}-\frac{c_2}{4}-{\lambda }^3.$$

In order to calculate the derivative of the real part of the eigenvalues of the Jacobian matrix we make use of Lemma 0.3. It is easy to see that in this case:

$$\bar{\mathbf{p}}=(\frac{1}{79}(-4)\sqrt{71+20i\sqrt{3}},\frac{16}{-3\sqrt{3}+17i},\frac{32}{17\sqrt{3}+9i})\mathbf{q}=(\frac{1}{8}(-3)(\sqrt{3}+i),\frac{5i}{8},\frac{\sqrt{3}}{8})\frac{dM}{d{c}_2}=\left(\begin{array}{ccc}\hfill 0\hfill & \hfill -2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill \end{array}\right)$$

A direct calculation shows the result. □

We now state the main result of this section which is a summary of the previous results.

Theorem 3.4

Under the hypothesis of the Proposition 3.3 the system (2) presents a supercritical Hopf bifurcation with respect to the parameter $c_2$ and bifurcation value $c_2=\frac{3}{5}$. Furthermore the first Lyapunov coefficient of the system with $c_2=\frac{3}{5}$ is ${\ell }_1(p_0,c_2)=-\frac{81495\sqrt{3}}{96064}$ and has a stable limit cycle for values $c_2>\frac{3}{5}$.

Proof

Proposition 3.3 shows that the system has a Hopf bifurcation with bifurcation parameter $c_2$ and with positive transversality condition. We only need to show that ${\ell }_1(p_0,c_2)$ has the required value. As this is only a matter of substituting the values into the formula given by Kuznetsov, we only show the elements involved in said formula.

Translating the equilibrium point $p_0$ to the origin. Taking $c_2=\frac{3}{5}$ we evaluate the system (12) at ${\tilde{p}}_0=(x+x_0,y+y_0,z+z_0)$, which produces the system

$$\dot{x}=-\frac{(x+1)(5x^3+5x^2+24xy+10x+24y)}{10(x^2+2x+2)},\dot{y}=\frac{(12y+5)(9x^{2}y-10x^{2}z+5x^2+18xy-20xz+10x+6y-20z)}{10(x^2+2x+2)(6y+5)},\dot{z}=\frac{3y(4z+1)}{2(6y+5)}.$$ (12)

The Jacobian matrix of corresponding to system (12) which is evaluated at the origin $0\in {\mathbb{R}}^3$ is

$$A=\left(\begin{array}{ccc}\hfill -\frac{1}{2}\hfill & \hfill -\frac{6}{5}\hfill & \hfill 0\hfill \\ \hfill \frac{1}{2}\hfill & \hfill \frac{3}{10}\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill \frac{3}{10}\hfill & \hfill 0\hfill \end{array}\right)$$

By direct calculations we obtain that the bilinear form B at the equilibrium $p=(0,0,0)$ is the bilinear form given by $B((x,y,z),(u,v,w))=(B_1,B_2,B_3)$, where

$$\begin{gathered} B_1=-\frac{6}{5}(uy+vx)-\frac{ux}{2} \\ {B}_2=\frac{1}{50}(60(uy+vx)-25ux-60(vz+wy)+36vy)B_3=\frac{6}{25}(-3vy+5vz+5wy) \end{gathered}$$

On the other hand, the trilinear form C at 0 acting on arbitrary vectors $(x,y,z),(u,v,w),(r,s,t)$ providing a vector $(C_{11},C_{12},C_{13})$ where

$$\begin{array}{c}{C}_{11}=\frac{6}{5}(ruy+rvx+sux),\hfill \\ C_{12}=\frac{6}{125}(-25(ruy+rvx+sux)+60(svz+swy+tvy)-54svy),\hfill \\ C_{13}=\frac{36}{125}(9svy-10(tvy+swy+svz)).\hfill \end{array}$$

Theorem 0.1 gives the first Lyapunov coefficient

$${\ell }_1(0)=Re\left(\frac{-81495\sqrt{3}-83711i}{96064}\right)=\frac{-81495\sqrt{3}}{96064}<0.$$

 □

According to the Theorem 3.4 we have a Hopf bifurcation at the equilibrium point $p_0$ with respect of the parameter $c_2$ and bifurcation value $c_{20}=\frac{3}{5}$. For values of $c_2$ less than $c_{20}$, $p_0$ is asymptotically stable, if $c_2$ is greater than $c_{20}$ the system (12) exhibits a stable limit cycle. This is illustrated in the bifurcation diagram shown in Fig. 1.

Figure 1.

Hopf bifurcation diagram at p₀.

3.2. Hopf bifurcation for the case $p_1$

In a similar way as we treated the equilibrium point $p_0$ we will show that an stable limit cycle appears near the equilibrium point $p_1$. Recall Corollary 2.3, the conditions given in this Corollary are summarized in the following table.

With these parameters then the three equilibrium points of the system which are in Ω are given by:

$$p_0=(x_0,\frac{c_1{x}_0(1+x_0)\rho }{2(c_2+2c_2{x}_0)},\frac{c_1{c}_3{x}_0(1+x_0)\rho }{2(d_2+2d_2{x}_0)}),$$ (13)

$$p_1=(\frac{x_0}{2},\frac{c_1{x}_0(1+x_0)\rho }{2(c_2+2c_2{x}_0)},\frac{c_1{c}_3{x}_0^2\rho }{4d_2+8d_2{x}_0}),$$ (14)

$$p_2=(\frac{2+x_0}{2},\frac{c_1{x}_0(1+x_0)\rho }{2(c_2+2c_2{x}_0)},\frac{c_1{c}_3{x}_0(4+x_0)\rho }{4(d_2+2d_2{x}_0)}),$$ (15)

and the system becomes:

$$\begin{gathered} \dot{x}=x(-\frac{6c_{2}x(3x_0+2)y}{c_1(x^2(8x_0+4)+x_0^2(x_0+2))}+\rho -\frac{\rho x}{2x_0+1}),\dot{y}=c_{2}y(-\frac{2d_2(2x_0+1)z(d_2+k_1)}{c_1{c}_3{k}_1\rho x_0(x_0+1)+2c_2{c}_3{d}_2(2x_0+1)y} \\ +\frac{6x^2(3x_0+2)}{x^2(8x_0+4)+x_0^2(x_0+2)}-1),\dot{z}=d_{2}z(\frac{2c_2(2x_0+1)y(d_2+k_1)}{c_1{k}_1\rho x_0(x_0+1)+2c_2{d}_2(2x_0+1)y}-1). \end{gathered}$$ (16)

The treatment of the point is very similar to the case of $p_0$ we omit the partial results and present only the main results, whose proofs are similar as the above section.

Proposition 3.5

For the value

$$c_2=\frac{2{\rho }^2(d_2+k_1)(d_2(x_0+1)(x_0(9x_0+19)+8)+k_1(x_0+2){(3x_0+2)}^2)}{3d_2{x}_0(d_2(x_0+1)(3k_1{x}_0(2x_0+1)+\rho (x_0(9x_0+19)+8))+k_1\rho (x_0+2){(3x_0+2)}^2)},$$

the eigenvalues of the Jacobian matrix at $p_1$ of the system (16) are:

$$\begin{gathered} \alpha =-\frac{d_2{k}_1\rho x_0(2x_0+1)}{d_2(x_0+1)(3k_1{x}_0(2x_0+1)+\rho (x_0(9x_0+19)+8))+k_1\rho (x_0+2){(3x_0+2)}^2}<0\mathit{\text{,}} \\ i\omega ,-i\omega , \end{gathered}$$

where $\omega =\frac{{\rho }^2(d_2(x_0+1)(x_0(9x_0+19)+8)+k_1(x_0+2){(3x_0+2)}^2)}{9d_2{x}_0{(x_0+1)}^2(2x_0+1)}>0$.

Theorem 3.6

There exists a set of parameters for which the system (2) presents a supercritical Hopf bifurcation at the point $p_1$ with respect to the parameter $c_2$.

Proof

Choose the following parameter values:

As we did in the case of $p_0$, we prove that the system (2) presents a Hopf bifurcation with respect to the parameter $c_2$ and bifurcation value $c_{20}=\frac{196}{165}$. The transversality value is $\frac{148225}{2372176}$, the first Lyapunov coefficient is $-\frac{200145731139991}{21636672597353}$ and then we have a stable limit cycle for values $c_2>\frac{196}{165}$. For values of $c_2$ less than $c_{20}$, $p_1$ is asymptotically stable. This is illustrated in the bifurcation diagram shown in Fig. 2. □

Figure 2.

Hopf bifurcation diagram at p₁.

3.3. Hopf bifurcation at $p_2$ and the existence of a stable limit cycle

As before we will only show the main result, showing the existence of a limit cycle near the equilibrium point $p_2$. Recall 3.2 and the three equilibrium points given by (13). With these assignations, we obtain the following results for the point $p_2$.

Proposition 3.7

If $k_1=d_2$ and $x_0=1$, then for the value

$$c_2=\frac{52{\rho }^2}{1250d_2+325\rho },$$

the eigenvalues of the Jacobian matrix at $p_2$ of the system (16) are:

$$\alpha =-\frac{5d_2\rho }{50d_2+13\rho }<0,i\omega ,-i\omega ,$$

where $\omega =\frac{13{\rho }^2}{500}>0$.

Theorem 3.8

There exists a set of parameters for which the system presents a supercritical Hopf bifurcation at $p_2$, the first Lyapunov coefficient is negative and hence the there exists a stable limit cycle near $p_2$.

Proof

Choose the following parameter values:

It is straightforward to show that the system (2) presents a supercritical Hopf bifurcation with respect to the parameter $c_2$ an bifurcation value $c_{20}=\frac{26}{64125}$. The transversality value is $\frac{17105985}{74739152}$, the first Lyapunov coefficient is $-\frac{20569384970086914}{719004931251198015875}\sqrt{\frac{13}{5}}$ and has a stable limit cycle for values $c_2>\frac{26}{64125}$. For values of $c_2$ less than $c_{20}$, $p_2$ is asymptotically stable. This is illustrated in the bifurcation diagram shown in Fig. 3. □

Figure 3.

Hopf bifurcation diagram at p₂.

4. Simultaneous Hopf bifurcation

Using the same tools as before we prove the existence of simultaneous Hopf bifurcation, at $p_0$ and $p_1$.

As previously done we will only show the main results, this is done to avoid long hard to read expressions and to simplify the presentation of the results.

Theorem 4.1

There exists a set of parameters for which the system (2) has a simultaneous Hopf bifurcation at $p_0$ and $p_1$.

Proof

If we choose the values of the parameters as:

As before, it is a straightforward calculation to show that the system presents a simultaneous Hopf bifurcation at $p_0$ and $p_1$, with respect to the parameter $c_2$ with bifurcation value

$$c_{20}=\frac{5985186238797318379008}{741250407958267590625}.$$

In this case:

$${\ell }_1(p_0)\approx 0.00378712\text{and}{\ell }_1(p_1)\approx 0.000594732,$$

with respective transversality coefficients:

$$\frac{1463296736069062273923850}{2955890437009175285513677}\text{and}\frac{1146011225593717693235471875}{5174577775374171114824921173}.$$

This shows that the system presents two unstable limit cycles at the points $p_0$ and $p_1$ which have two locally stable manifolds of dimension 2. Hence we have two regions of stability for the system. □

According to the previous calculus, for values of $c_2$ less than $c_{20}$ the equillibrium points $p_0$ and $p_1$ are locally stable and for values of $c_2$ greater than the parameter bifurcation value we obtain at least one unstable limit cycle.

5. Conclusion

When considering a tritrophic model where the interaction between the prey and the mesopredator has a functional response Holling III and in the predator-superpredator interaction a Holling II, we show parameter conditions for which the system (2) has up to three coexistence equilibrium points. In each of these three equilibrium points, we show that the system exhibits a supercritical Hopf bifurcation and the bifurcation parameter is the mortality rate $c_2$ of the mesopredator. Then there are different intervals where it is possible to take the mortality rate $c_2$ and guarantee the coexistence of the species. Moreover, the coexistence is given by a small amplitude limit cycle generated by a Hopf bifurcation. On the other hand, the system exhibits bistability, which is given through two stable equilibrium points each having a unstable limit cycle near them.

Declarations

Author contribution statement

G. Blé, V. Castellanos, F. E. Castillo-Santos: Contributed reagents, materials, analysis tools or data; Wrote the paper.

Funding statement

The third author was partially supported by CONACYT grant number CB-2014-243722 and the second author by grant number A1 - S - 47710.

Data availability statement

No data was used for the research described in the article.

Declaration of interests statement

The authors declare no conflict of interest.

Additional information

Supplementary content related to this article has been published online at https://doi.org/10.1016/j.heliyon.2021.e06212.

No additional information is available for this paper.

Supplementary material

The following Supplementary material is associated with this article:

AppendixA.pdf

Appendix A: Bistability and Hopf bifurcation of a tritrophic system with Holling functional responses.