Global dynamics of a prey-predator model with Allee effect and additional food for the predators

Abstract

Provision of additional/alternative food to the predators for controlling predator-prey dynamics has been receiving considerable attention from theoretical as well as experimental biologists. This is due to environment friendly role played by the additional food in controlling and managing the interacting population. Theoretical investigations done on additional food provided predator-prey models reveal that provision of additional food to predators has a significant role to play in enhancement of commercially important predator species and also in reduction of prey population. The quality and quantity of the additional food provided to predators play vital role in shaping the dynamics of the interacting system. So far as our knowledge goes, all theoretical investigations carried out in this direction assume logistic growth for the prey species. In reality, the per capita growth rate of the prey population is often an increasing function at low prey density. Incorporation of this realistic growth rate induces Allee effect into the dynamics of the prey species. In this paper, we consider an additional food provided predator-prey model wherein the prey population is subjected to Allee effect. The model includes both strong and weak Allee effects. This article presents a comprehensive analysis of the considered model that highlights the influence of Allee effect in prey and additional food for the predators on the system dynamics.

Appendix

Let ${{\mathbf {f}}}(x,y;\delta )$ represent the vector

$$\begin{aligned} {{\mathbf {f}}}(x,y; \delta ) = \left( \begin{array}{c} x(x - \theta )(1 - \frac{x}{\gamma }) - \frac{xy}{1 + \alpha \xi + x}\\ \frac{\beta (x+\xi )y}{1 + \alpha \xi + \gamma } - \delta y \\ \end{array} \right) . \end{aligned}$$

(25)

Differentiate the above function with respect to $\delta$, we obtain

$$\begin{aligned} {{\mathbf {f}}}_{\delta }(x,y; \delta ) = \left( \begin{array}{c} 0\\ -y\\ \end{array} \right) . \end{aligned}$$

(26)

The jacobian matrix of the prey-predator system (8)–(9) at the equilibrium point $E_{\theta }$ is given by

$$\begin{aligned} P = J_1(\theta , 0) = \left( \begin{array}{cc} \frac{\theta }{\gamma }(\gamma - \theta ) &{} -\frac{\theta }{1 + \alpha \xi + \theta } \\ 0 &{} \frac{\beta (\theta +\xi )}{1 + \alpha \xi +\theta } - \delta \\ \end{array} \right) . \end{aligned}$$

Observe that the Jacobian matrix P has a simple zero eigen values as its determinant is zero and the trace is different from zero whenever $x_* = \theta$. Let $\mathbf{v } = \left( \begin{array}{c} v_1 \\ v_2 \\ \end{array} \right)$ be an eigen vector of the Jacobian matrix P corresponding to the eigen value $\lambda = 0$. We have

$$\begin{aligned} P\mathbf{v }= & {} \left( \begin{array}{cc} \frac{\theta }{\gamma }(\gamma - \theta ) &{}\quad -\frac{\theta }{1 + \alpha \xi + \theta } \\ 0 &{} 0 \\ \end{array} \right) \left( \begin{array}{c} v_1 \\ v_2 \\ \end{array} \right) \\= & {} \left( \begin{array}{c} \frac{\theta }{\gamma }(\gamma - \theta )\quad v_1 - \frac{\theta v_2}{1 + \alpha \xi + \theta } \\ 0 \\ \end{array} \right) \end{aligned}$$

and

$$\begin{aligned} \lambda \mathbf{v } = \left( \begin{array}{c} \lambda v_1 \\ \lambda v_2 \\ \end{array} \right) = \left( \begin{array}{c} 0 \\ 0 \\ \end{array} \right) . \end{aligned}$$

We know that the vector $\mathbf{v }$ will be an eigen vector of P corresponding to the eigen value $\lambda = 0$ if $P\mathbf{v } = \lambda \mathbf{v }$. Clearly the components of this eigen vector satisfy the equation

$$\begin{aligned} \frac{\theta }{\gamma }(\gamma - \theta )v_1 - \frac{\theta v_2}{1 + \alpha \xi + \theta } = 0 \end{aligned}$$

By choosing $v_1 = 1$ we obtain an eigen vector for P given by

$$\begin{aligned} \mathbf{v } = \left( \begin{array}{c} 1 \\ \frac{(\gamma - \theta )(1 + \alpha \xi + \theta )}{\gamma } \\ \end{array} \right) . \end{aligned}$$

(27)

Now, let $\mathbf{w } = \left( \begin{array}{c} w_1 \\ w_2 \\ \end{array} \right)$ be the eigen vector of the matrix $P^{\top }$ then

$$\begin{aligned} P^{\top } = \left( \begin{array}{cc} \frac{\theta }{\gamma }(\gamma - \theta ) &{} 0 \\ \frac{\theta }{1 + \alpha \xi + \theta } &{} 0 \\ \end{array} \right) \left( \begin{array}{c} w_1 \\ w_2 \\ \end{array} \right) = \left( \begin{array}{c} \frac{\theta }{\gamma }(\gamma - \theta )w_1 \\ \frac{\theta w_1}{1 + \alpha \xi + \theta } \\ \end{array} \right) \end{aligned}$$

and

$$\begin{aligned} \lambda \mathbf{w } = \left( \begin{array}{c} \lambda w_1 \\ \lambda w_2 \\ \end{array} \right) . \end{aligned}$$

Considering $\lambda \mathbf{w } = P^{\top }w$ we obtain the equations

$$\begin{aligned} \lambda w_1= & {} \frac{\theta }{\gamma }(\gamma - \theta )w_1,\\ \lambda w_2= & {} \frac{\theta w_1}{1 + \alpha \xi + \theta }. \end{aligned}$$

Now $\lambda = 0$ implies that $w_1 = 0$ wince $\gamma - \theta \ne 0$ and $w_2$ is arbitrary. Hence eigen vector of $P^{\top }$ corresponding the eigen value $\lambda = 0$ is given by

$$\begin{aligned} \mathbf{w } = \left( \begin{array}{c} 0 \\ 1 \\ \end{array} \right) = \left( \begin{array}{c} w_1 \\ w_2 \\ \end{array} \right) . \end{aligned}$$

From Eq. (26) we have

$$\begin{aligned} {{\mathbf {f}}}_{\delta }(\theta , 0; \delta ) = \left( \begin{array}{c} 0 \\ 0 \\ \end{array} \right) \end{aligned}$$

and hence

$$\begin{aligned} \mathbf{w }^{\top }{{\mathbf {f}}}_{\delta }(\theta , 0; \delta ) = \left( \begin{array}{cc} 0 &{} 1 \\ \end{array} \right) \left( \begin{array}{c} 0 \\ 0 \\ \end{array} \right) = 0. \end{aligned}$$

(28)

Now, let us consider

(29)

The expansion of the term $D^2{{\mathbf {f}}}(x,y;\delta )( \mathbf{v } , \mathbf{v })$ is given by

(30)

Evaluating $D\, {{\mathbf {f}}}_\delta (x,y; \delta ) \mathbf{v }$ at $(x,y) = (\theta , 0)$ using (27) and (29) we obtain

$$\begin{aligned} D\, {{{\mathbf {f}}}_\delta } (x,y; \delta ) \mathbf{v }&= \left( \begin{array}{c} 0 \\ 0 \\ \end{array} \right) .1 + \left( \begin{array}{c} 0\\ -1 \\ \end{array} \right) \frac{(1 + \alpha \xi + \theta )(\gamma - \theta )}{\gamma }\\&= \left( \begin{array}{c} 0 \\ -\frac{(1 + \alpha \xi + \theta )(\gamma - \theta )}{\gamma } \\ \end{array} \right) \end{aligned}$$

and hence we have

$$\begin{aligned} \mathbf{w }^{\top } [D\, {{\mathbf {f}}}_\delta (\theta , 0; \delta ) \mathbf{v }]&= \left( \begin{array}{cc} 0 &{} 1 \\ \end{array} \right) \left( \begin{array}{c} 0 \\ -\frac{(1 + \alpha \xi + \theta )(\gamma - \theta )}{\gamma } \\ \end{array} \right) \nonumber \\&= -\frac{(1 + \alpha \xi + \theta )(\gamma - \theta )}{\gamma } \ne 0. \end{aligned}$$

(31)

Evaluating $\mathbf{w }^{\top }D^2 {{\mathbf {f}}}(\theta ,0;\delta )(\mathbf{v }, \mathbf{v })$ using (27) and (30) we obtain

$$\begin{aligned}&\mathbf{w }^{\top } [D^2 {{\mathbf {f}}}(\theta , 0; \delta )( \mathbf{v } , \mathbf{v })]\nonumber \\&\quad = \left( \begin{array}{cc} 0 &{} 1 \\ \end{array} \right) \left( \begin{array}{c} -6\frac{\theta }{\gamma } + 2\left( 1 + \frac{\theta }{\gamma }\right) + \frac{2(1 + \alpha \xi )}{(1 + \alpha \xi + \theta )^3} \\ 2\frac{\beta [(1+\alpha \xi ) - \xi ](\gamma - \theta )}{(1 + \alpha \xi + \theta )}\\ \end{array} \right) \nonumber \\&\quad = 2\frac{\beta [(1+\alpha \xi ) - \xi ](\gamma - \theta )}{(1 + \alpha \xi + \theta )} \ne 0. \end{aligned}$$

(32)

From Sotomayer theorem [46] together with the Eqs. (28), (31) and (32), the prey-predator system (8)–(9) experiences transcritical Bifurcation at $E_{\theta }$. In a similar manner, we can also prove that the prey-predator system (8)–(9) experiences transcritical Bifurcation at $E_{\gamma }$