A Three Species Food Chain Model with Fear Induced Trophic Cascade

Abstract

In ecology, predator–prey interaction is one of the most important factors. The effects of predators on prey population can be direct and deadly, or it may be indirect and non-consumptive. Recent experimental findings have explored that fear of predator (indirect effect) alone can change prey’s behavior including reproduction and foraging. Suraci et al. (Nature Communications, 7, 10698, 2016) experimentally showed that fear of large carnivore reduces mesocarnivore foraging, which benefits the mesocarnivore’s prey. They also showed that fear of large carnivore mediates a cascading effect in lower trophic level. In the present study, our aim is to observe how the cascading effects of fear in a tri-trophic food chain model influence the dynamics of the model. We propose a three-species food chain model incorporating the cost of fear into the predation rate of middle predator. We consider the fact that due to fear of the top predator, middle predator forage less. As a result, the predation rate of middle predator decreases which reduces the predation pressure on basal prey. Mathematical properties such as boundedness, persistence, equilibria analysis, local and global stability analysis of the model are investigated. We perform bifurcation analysis around interior equilibrium point of the system. We notice that cost of the fear in middle predator can stabilize an otherwise chaotic system. We also investigate the robustness of the stabilizing role of the fear parameter. We observe that system initiating from the different dynamical regime, fear ultimately drives the system towards stability. It is also found that for increasing the level of fear, the system enters into a stable state through multiple switching of dynamics. Our results suggest that cost of the fear in middle predator can stabilize the system and enhances persistence of the system. We illustrate our analytical results numerically. Finally our results qualitatively reflect the experimental findings of Suraci et al.

Appendix

Positivity, Boundedness and Permanence of the system

Positivity

Since the state variables of the system (3) represent population densities, positivity implies that population never become zero and always survive.

Theorem 7.1

All solutions of the system (3) having positive initial values remain positive.

Proof

Let ${R^3_+=[0, \infty )^3}$ be the nonnegative octant in ${R^3}$. The right hand side of the system (3) are continuously differentiable and satisfy locally Lipschitz conditions. So, the solution of the system (3) with nonnegative initial conditions satisfies non negativity condition and uniquely exists in [0, M), where $M (>0)$ is a sufficiently large number ([39] Theorem A.4). $\square$

Boundedness

Boundedness of the predator–prey system means that due to limited resource none of the interacting species of the system grows abruptly or exponentially for a long-time interval.

Theorem 7.2

All the solutions of the system (3) having positive initial conditions in $R^3_{+}-\{0\}$ are bounded.

Proof

We define a function P which is given by, $P=x+y+z$.

Differentiating the above equation w.r.t. time along with the solution of (3) is

$$\begin{aligned} \frac{dP}{dt} =\frac{dx}{dt}+\frac{dy}{dt}+\frac{dz}{dt} =x(1-x)-d_1y-d_2z. \end{aligned}$$

Therefore, $\frac{dP}{dt}+\mu P =x(1-x+\mu )-(d_1-\mu )y-(d_2-\mu )z\le \frac{(1+\mu )^2}{4}=Q$(say), where $\mu \le min\{d_1,d_2\}$.

We obtain the following inequality by using standard theory of differential inequality

$$\begin{aligned} 0\le P\le \frac{Q(1-e^{-\mu t})}{\mu }+P(x(0),y(0),z(0))e^{-\mu t} . \end{aligned}$$

Therefore, as $t\rightarrow \infty$ we get $0\le P\le \frac{Q}{\mu }$.

So, all the solutions of system (3) in $R^3_{+}-\{0\}$, having positive initial conditions are bounded in the region $W=\{(x,y,z)\epsilon R^3_+:P\le \frac{Q}{\mu }+\eta$, for any $\eta >0$ }. $\square$

Permanence

Persistence of a system implies that the minimal densities of all population are away from zero and bounded. Also all population can coexist for long range of time [16].

Theorem 7.3

Let $r_1$, $r_2$, $r_3$ , $R_1$, $R_2$, $R_3$ are positive constants does not dependent on the initial conditions of system (3). If the subsequent inequalities holds

$$\begin{aligned} r_1\le & {} \displaystyle \lim _{t\rightarrow \infty }inf\ x(t)\le \displaystyle \lim _{t\rightarrow \infty }sup\ x(t)\le R_1 \\ r_2\le & {} \displaystyle \lim _{t\rightarrow \infty }inf\ y(t)\le \displaystyle \lim _{t\rightarrow \infty }sup\ y(t)\le R_2 \\ r_3\le & {} \displaystyle \lim _{t\rightarrow \infty }inf\ z(t)\le \displaystyle \lim _{t\rightarrow \infty }sup\ z(t)\le R_3 \end{aligned}$$

with initial conditions $x(0)>0$, $y(0)>0$, $z(0)>0$, then we say the system (3) is permanent.

Proof

From system (3) we obtain,

$$\begin{aligned} \frac{dx}{dt}\le & {} x(1-x),\nonumber \\ \frac{dy}{dt}\le & {} y\left( a_1x-\frac{a_2z}{1+b_2y}-d_1\right) ,\nonumber \\ \frac{dz}{dt}\le & {} z(a_2y-d_2). \end{aligned}$$

(17)

Now applying the standard comparison theorem [44], we get

$$\begin{aligned} \displaystyle \lim _{t\rightarrow \infty }sup\ x(t)\le R_1 , \displaystyle \lim _{t\rightarrow \infty }sup\ y(t)\le R_2 , \displaystyle \lim _{t\rightarrow \infty }sup\ z(t)\le R_3 \end{aligned}$$

(18)

where $R_1=1$, $R_2=\frac{d_2}{a_2}$, $R_3=\frac{(a_1-d_1)(a_2+b_2d_2)}{a_2^2}$.

Here $R_1$, $R_2$ are always positive and $R_3$ is positive if $a_1>d_1$. Similarly, from equations (3), we obtain

$$\begin{aligned} \frac{dx}{dt}\ge & {} [(1-x)-a_1y]x,\nonumber \\ \frac{dy}{dt}\ge & {} \left[ \frac{a_1x}{1+b_1x}\frac{1}{1+kz}-a_2z-d_1\right] y,\nonumber \\ \frac{dz}{dt}\ge & {} \left[ \frac{a_2y}{1+b_2R_2}-d_2\right] z. \end{aligned}$$

(19)

Similarly using the standard comparison theorem [44], we have

$$\begin{aligned} \displaystyle \lim _{t\rightarrow \infty }inf\ x(t)\ge r_1 , \displaystyle \lim _{t\rightarrow \infty }inf\ y(t)\ge r_2 , \displaystyle \lim _{t\rightarrow \infty }inf\ z(t)\ge r_3 \end{aligned}$$

(20)

where $r_1=\frac{a_2-a_1d_2(1+b_2R_2)}{a_2}$, $r_2=\frac{d_2(1+b_2R_2)}{a_2}$ and

$$\begin{aligned} r_3=\frac{-(a_2+d_1k)+\sqrt{(a_2+d_1k)^2-4a_2k(d_1-\frac{a_1a_2-a_1^2d_2(1+b_2R_2)}{a_2+b_1a_2-a_1b_1d_2(1+b_2R_2)}})}{2a_2k}. \end{aligned}$$

Here, $r_1, r_2, r_3$ are positive if $d_1<\frac{a_1a_2-a_1^2d_2(1+b_2R_2)}{a_2+b_1a_2-a_1b_1d_2(1+b_2R_2)}$ and $d_2<\frac{a_2}{a_1(1+b_2R_2)}.$

From Eqs. (18) and (20) we conclude that the system (3) is permanent. $\square$