The influence of fear effect to the Lotka–Volterra predator–prey system with predator has other food resource

A Lotka–Volterra predator–prey system incorporating fear effect of the prey and the predator has other food resource is proposed and studied in this paper. It is shown that the trivial equilibrium and the predator free equilibrium are both unstable, and depending on some inequalities, the system may have a globally asymptotically stable prey free equilibrium or positive equilibrium. Our study shows the fear effect is one of the most important factors that lead to the extinction of the prey species. Such a finding is quite different from the known result. Numeric simulations are carried out to show the feasibility of the main results.


Introduction
The aim of this paper is to investigate the dynamic behaviors of the following Lotka-Volterra predator-prey system incorporating fear effect of the prey and the predator having other food resource: where u and v are the density of prey species and the predator species at time t, respectively. r 0 is the birth rate of the prey species, d is the death rate of the prey species, a is the intraspecific competition of the prey species, m is the intrinsic grow rate of the predator species; p denotes the strength of interspecific between prey and predator; c is the conversion efficiency of ingested prey into new predators; d 1 is the intraspecific competition of the predator species; k is the level of fear, which is due to anti-predator behaviors of the prey. Here we make the assumption that without the prey species, the predator species satisfies the logistic equation It is well known that the logistic equation admits a unique positive equilibrium which is globally attractive. Therefore, here we assume that the predator species could be permanent without the affordable of the prey species. Such an assumption means that the predator species takes other species as food resources. The predator-prey relationship has been highly valued by scholars because of its widespread existence . Wang, Zanette and Zou [1] proposed the following Lotka-Volterra predator-prey system incorporating the fear effect of the prey: (1.3) The system admits three nonnegative equilibria, E 0 (0, 0), Concerned with the dynamic behaviors of the system (1.3), the authors obtained the following result (see Theorems 3.1 and 3.2 in [1]).
Theorem A Assume that r 0 < d, then E 0 is globally asymptotically stable; The boundary equilibrium E 1 is globally asymptotically stable if r 0 ∈ (d, d + am cp ), and the unique positive equilibrium E 2 is globally asymptotically stable if r 0 > d + am cp .
One could easily see that in Theorem A, all the conditions are independent of k, which means that the fear effect of the prey species has no influence to the dynamic behaviors of the system. It brings to our attention that in system (1.3), without the prey species, the predator species satisfies the equation This indicates that in system (1.3), the predator species has prey as its unique food resource. Now, it is well known that in the nature, predator species often take many species as its food resource, such that if one resource is scarce, it could take other food resource to maintain its life. This leads us to propose the system (1.1). The aim of this paper is to investigate the dynamic behaviors of the system (1.1), and to find the influence of the fear effect on the prey species.
The rest of the paper is arranged as follows. We will investigate the local and global stability property of the equilibria of the system (1.1) in Sects. 2 and 3, respectively, and then discuss the influence of fear effect in Sect. 4. Numeric simulations are presented in Sect. 5 to show the feasibility of the main results. We end this paper with a brief discussion.

The existence and local stability of the equilibria of system (1.1)
Concerned with the existence of the equilibria of system (1.1), we have the following results.
Theorem 2.1 System (1.1) always admits the trivial boundary equilibrium E 0 (0, 0) and prey free equilibrium E 1 (0, m d 1 ), if r 0 > d holds, then the predator free equilibrium E 2 ( r 0 -d a , 0) exists. Also, there exists a unique positive equilibrium E 3 and u * is the unique positive solution of the equation where A 1 = c 2 kp 3 + acd 1 kp, Proof The equilibrium of system (1.1) satisfies the equation From the second equation of (2.4), one has v = 0 or v = cpu+m d 1 .
This ends the proof of Theorem 2.1.
Remark 2.1 One may curiously, whether the system could exist two, one or none positive equilibrium, since this may lead to the saddle-node bifurcation, however, this is impossible. Indeed, if A 3 ≥ 0, let us consider the function should intersect with the negative half part of the u-axis, which means that the other one solution of Eq. (2.6) should be negative. Therefore, it is enough for us to consider the situation of (2.1) and need not to investigate the case holds, the predator free equilibrium E 2 ( r 0 -d a , 0) is unstable, and the prey free equilibrium holds, i.e., the positive equilibrium is locally asymptotically stable as long as it exists.
Proof The Jacobian matrix of the system (1.1) is calculated as (2.14) Then the Jacobian matrix of the system (1.1) about the trivial equilibrium E 0 (0, 0) is The eigenvalues of J(E 0 ) are λ 1 = r 0d, λ 2 = m > 0. Thus, the trivial equilibrium E 0 (0, 0) is unstable since one of the eigenvalues is positive. It follows from (2.3) that the Jacobian matrix of the system (1.1) about the predator free equilibrium E 2 (u, 0), where u = r 0 -d a , is Noting that (u * , v * ) satisfies the equation Then the Jacobian matrix of the system (1.1) about the positive equilibrium E 3 Then we have So both eigenvalues of J(E 3 (u * , v * )) have negative real parts, consequently, E 3 (u * , v * ) is locally asymptotically stable.
This ends the proof of Theorem 2.2.

Global asymptotical stability
The aim of this section is to investigate the global stability property of the prey free equilibrium E 2 (0, m d 1 ) and the positive equilibrium E 3 (u * , v * ) of system (1.1). Indeed, we have the following result.

Theorem 3.1
(i) The prey free equilibrium E 1 (0, m d 1 ) is globally asymptotically stable if holds, i.e., the positive equilibrium is globally asymptotically stable as long as it exists.
Proof (i) We will prove (i) by constructing some suitable Lyapunov function. Let us define a Lyapunov function where v = m d 1 .
Then the time derivative of V 1 along the trajectories of (1.1) is Thus, V 1 (x, y) satisfies the Lyapunov asymptotic stability theorem, and the boundary equilibrium E 1 (0, m d 1 ) of system (1.1) is globally asymptotically stable. (ii) Under the assumption of (3.2), it follows from Theorem 2.2 that system (1.1) admits a unique positive equilibrium, which is locally asymptotically stable, and the prey free equilibrium is unstable. To show that E 3 (u * , v * ) is globally asymptotically stable, it is enough to show that the system admits no limit cycle in the first quadrant (see [25][26][27][28][29][30][31][32][33][34]). Let us consider the Dulac function B(u, v) = 1 uv , then (3.6) By the Dulac theorem [34], there is no closed orbit in the first quadrant. Consequently, E 3 (u * , v * ) is globally asymptotically stable. The proof of Theorem 3.1 is finished.

The influence of fear effect
In the following we will discuss the effect of the fear effect. Denote By simple computation, we have for all u * > 0, v * > 0, k > 0. Thus, Eqs. (4.2) satisfy the conditions of the existence theorem for implicit functions, then Eqs. (4.2) determine the two implicit functions of for all k > 0. Also, By computation, we have and so du * dk < 0, dv * dk < 0, that is, both the prey and predator density are a decreasing function of k.

Numeric simulations
We will introduce two examples to show the feasibility of the main results.
then one could see that Hence, it follows from Theorem 3.1 that the prey free equilibrium E 2 (0, 1) of system (5.1) is globally asymptotically stable. Numeric simulation (Fig. 1) also supports this assertion.
Example 5.2 Let us consider the following model: Here, corresponding to system (1.1), we take r 0 = 5, d = 1, k = a = p = q 1 = m = q 2 = 1, c = 0.5, then one could see that  Now, based on system (5.3), let us furthermore consider the system (5.5) Obviously, the positive equilibrium E 3 (u * , v * ) satisfies the equation Solving (5.6) we obtain u * = -5k -3 + √ k 2 + 66k + 9 3k , v * = 1 2 u * + 1. (5.7) From Fig. 4, one could see that u * is the decreasing function of k. This is in coincidence with the analysis result of the previous section. Also, from the relationship of v * and u * , one could easily see that v * is a decreasing function of k.

Conclusion
Wang, Zanette and Zou [6] proposed a Lotka-Volterra predator-prey system incorporating the fear effect of prey species, i.e., system (1.3). Their result (Theorem A) indicates that the fear effect has no influence to the existence and stability of the equilibria. That is, if for the system without fear effect there exists a positive equilibrium, then the system with fear effect also admits a unique positive equilibrium, which is globally asymptotically stable. Hence, the fear effect of prey species has no influence on the persistent property of the system. Stimulated by the fact that the predator species generally speaking is omnivorous, and if a particular kind of food source die out, they will feed on other food resources, we propose the system (1.1). One may conjecture that system (1.1) has similar dynamic behaviors to that of system (1.3). However, our study shows that the trivial equilibrium and the predator free equilibrium are both unstable, while the prey free equilibrium may be globally asymptotically stable. This affirms the fact that the predator species may still be permanent, despite the extinction of the prey species.
There are also some similarities between the system (1.1) and (1.3), indeed, if the positive equilibrium of system (1.1) exists, it is globally asymptotically stable, this is similar to the property of system (1.1).
From the numeric simulation (Fig. 4) of Example 5.2, one could see that with increasing fear effect, the final density of the prey species may approach zero, which means that the prey species will finally be driven to extinction. That is, the fear phenomenon has a negative effect on the survival of the prey species, it may be one of the essential factor that leads to the extinction of prey species. Such a finding is quite different from the result of Wang, Zanette and Zou [6].