The Influence of Fear Effect to a Discrete-Time Predator-Prey System with Predator Has Other Food Resource

A discrete-time predator–prey system incorporating fear effect of the prey with the predator has other food resource is proposed in this paper. The trivial equilibrium and the predator free equilibrium are both unstable. A set of sufficient conditions for the global attractivity of prey free equilibrium and interior equilibrium are established by using iteration scheme and the comparison principle of difference equations. Our study shows that due to the fear of predation, the prey species will be driven to extinction while the predator species tends to be stable since it has other food resource, i.e., the prey free equilibrium may be globally stable under some suitable conditions. Numeric simulations are provided to illustrate the feasibility of the main results.


Introduction
In ecology, predator-prey relationship is one of the most important relationship. The interaction between predator and prey is the core of evolutionary biology. In the development of biomathematics, many scholars have studied the dynamic behaviors of predators and prey in continuous models. However, in the past few decades, some scholars have also argued that discrete models are more appropriate than continuous ones when the populations have non-overlapping generations [1,2]. Scholars investigated the persistence, periodic solutions, probabilistic periodic solutions, etc. of non-autonomous discrete systems [3][4][5][6][7]. In addition, many scholars have successively studied the dynamical behaviors of autonomous discrete-time systems [8][9][10][11][12][13][14][15]. For example, Chen and Teng [8] discussed the stability of positive equilibrium of a two-species discrete competition system. Chen [9] investigated the global attractivity property of a discrete competition model. In [10], a discrete May type cooperative model incorporating Michaelis-Menten type harvesting is studied by Zhu et al. Stability, bifurcation and chaos of discrete predator-prey systems are investigated in [11][12][13][14][15]. The above studies show that discrete systems often have more complex dynamic behaviors than the corresponding continuous ones.
However, most of the discrete models mentioned above only considered the case where the predator kills the prey directly. In recent years, scholars have gradually noting that besides the direct hunting, the presence of predators would make the prey afraid, which affects their habitat, reproduction rate, foraging behavior, etc. Further, some scholars pointed out that fear effect has an influence on prey populations even more powerful than direct hunting [16][17][18][19][20][21][22][23][24][25][26][27][28][29].
In [30], Wang, Zanette and Zou obtained that the fear effect has no influence on the dynamic behaviors of the system with linear functional response. Both predator and prey species will go to extinction if r 0 < d.
Discrete systems usually exhibit more complex dynamics than their continuous counterparts. To investigate the impact of fear effect in discrete predator-prey system, Kundu, Pal, Samanta, et al. [31] have discretized the continuous model proposed by Wang, which leads to the following system: x(n + 1) = x(n) exp[ r 0 1 + ky(n) − d − ax(n) − py(n)], y(n + 1) = y(n) exp[cpx(n) − m]. (2) In [31], the authors observed that system (2) becomes stable from chaotic dynamics when the fear parameter k increasing, which is quite different from the property of system (1). The results of [31] illustrated that discrete systems do have more complex dynamical behaviors.
On the other hand, Wang, Zanette and Zou in [30] found that predators and prey would become extinct simultaneously under some suitable conditions. The reason is that the predator species take prey species as its unique food resource, however, generally speaking, predators are omnivorous and do not hunt for only one kind of prey, they look for other food sources to ensure their survival if one of food source is scare. Thus, Zhu, Wu, Lai, et al. [32] proposed the continuous system (3): In [32], Zhu et al. showed that the trivial equilibrium and the predator free equilibrium are both unstable. And the following results are obtained (see Theorems 2.2 and 3.1 in [32]): Theorem 1. The prey free equilibrium is globally asymptotically stable if The positive equilibrium of system (3) is globally asymptotically stable if That is, the prey free equilibrium is globally asymptotically stable under some suitable condition. This affirms the fact that predators may still be permanent even if the prey populations become extinct, which differs from the result of Wang, Zanette and Zou.
To further analyze the complex dynamical behaviors of the discrete system, corresponding to the continuous system (3) we propose the following system: where x(n), y(n) are the population density of prey and predator at n-generation, respectively. r 0 , d denote the birth rate and natural death rate of the prey, respectively. a, h represent the death rate due to intraspecific competition of the prey and the predator respectively. b is the maximum predation rate of the predator, m denotes the intrinsic grow rate of the predator, c is the conversion rate from prey's bioenergy to predators' bioenergy, k describes the level of the fear effect.
The rest of the paper is arranged as follows. We investigate the existence and local stability of the equilibria in Sections 2 and 3, respectively. In Sections 4 and 5, some sufficient conditions to guarantee the global stability of the interior equilibrium point and prey free equilibrium are given, respectively. In Section 6, some numerical simulations are provided to show the feasibility of the main results in this paper. Finally, a brief conclusion is presented in Section 7.

The Existence of Equilibria
To determine the equilibrium of system (4), we solve the following equations: By simple computation, we obtain the following results.
Theorem 2. System (4) always has the trivial equilibrium E 0 (0, 0) and the prey free equilibrium holds, where y * = m + cbx * h and x * is the unique positive solution of the equation with A 1 = b 3 c 2 k + abchk, A 2 = 2b 2 ckm + bcdhk + ahkm + b 2 ch + ah 2 , Proof. The positive equilibrium of system (4) satisfies the equations We get y = m + cbx h from the second equation. Substituting y to the first equation of (9) leads to where A 1 , A 2 , A 3 is defined in (8).
Obviously, A 1 and A 2 are both positive, so the axis of symmetry of Equation (10) is on the left half axel. Under the assumption of (6), one could easily see that A 3 < 0, hence, (10) has a unique positive solution x * . Consequently, system (4) has a unique positive This ends the proof of Theorem 3.
Proof. The Jacobian matrix of the system (4) about The eigenvalues of J( the proof in Theorem 4, the above results can be easily obtained. Theorem 6. If r 0 > d, system (4) has the predator free equilibrium E 2 Proof. The Jacobian matrix of the system (4) about E 2 is The eigenvalues of J( A similar method of proof as Theorem 4, the above results can be easily obtained. (2) source if T ≤ 2 and S > T, or T > 2 and S > max(T, 2T − 4); Proof. Calculating Jacobian matrix J(E * ) of system (4) at E * , we have So the characteristic equation of the above matrix can be written as By simple computation, we obtain According to Lemmas 1 and 2, E * is a sink and it's stable, if the conclusion (1) of Theorem 7 holds.
Next, if the other conditions of Theorem 7 hold separately, E * is a source, saddle and non-hyperbolic, respectively, at which E * is unstable.
This completes the proof.

Global Stability of Interior Equilibrium
In this section, we discuss the global stability of interior equilibrium E * (x * , y * ) by using the method of iteration scheme. Firstly, we introduce the following lemmas: . Assume that sequence {u n } satisfy u n+1 = u n exp(α − βu n ),n ∈ N, where α and β are positive constants and u(0) > 0. Then

Lemma 5 ([33]).
Suppose that functions f , g : x)) for n ∈ Z + and x ∈ [0, ∞) and g(n, x) is nondecreasing with respect to x. If {x(n)} and {u(n)} are the solutions of the following difference equations x(n + 1) = f (n, x(n)), u(n + 1) = g(n, u(n)), respectively, and x(0) ≤ u(0)(x(0) ≥ u(0)), then x(n) ≤ u(n) (x(n) ≥ u(n)), f or all n ≥ 0. where Then equilibrium E * is globally attractive, that is, Proof. Let (x(n), y(n)) be any positive solution of system (4) with x(0) > 0 and y(0) > 0. Let We now claim that For the first iteration, we aim to show that U 1 ≤ M x 1 and U 2 ≤ M y 1 . From the first equation of system (4) we obtain Consider the auxiliary equation Since r 0 ≤ 1, by conclusion (ii) of Lemma 4, we have u(n) ≤ 1 a for all n ≥ 2, where is nondecreasing for u ∈ (0, 1 a ], from Lemma 5, we have x(n) ≤ u(n) for all n ≥ 2, where u(n) is the solution of system (4) with u(2) = x(2). By conclusion (i) of Lemma 4, we have Hence, for any sufficiently small > 0, there exists an integer From the second equation of system (4) we have Consider the auxiliary equation Since (18) holds, then we have m + cbM x 1 ≤ 1. By conclusion (ii) of Lemma 4, we have , from Lemma 5 we have y(n) ≤ u(n) for all n ≥ N 1 . By conclusion (i) of Lemma 4, we have Hence, for any sufficiently small > 0, there exists an integer For the second iteration, we aim to show that V 1 ≥ N x 1 and V 2 ≥ N y 1 . From the first equation of system (4) we have Consider the auxiliary equation Since (19) holds, we have we have x(n) ≥ u(n) for all n ≥ N 2 . By conclusion (i) of Lemma 4, we have Hence, for any sufficiently small > 0, there exists an integer N 3 > N 2 such that if From the second equation of system (4), we further have Since m + cbN x 1 ≤ m + cbM x 1 ≤ 1, a similar argument as above, we can obtain Hence, for any sufficiently small > 0, there exists an integer For the third iteration, we aim to show that From the first equation of system (4) for n ≥ N 4 we have Since M y 1 > N y 1 , we get A similar argument as above, we can obtain Hence, for any sufficiently small > 0, there exists an integer N 5 > N 4 such that if From the second equation of system (4) for n ≥ N 5 , we have Since m + cbM x 2 ≤ m + cbM x 1 ≤ 1, a similar argument as above, we can obtain Hence, for any sufficiently small > 0, there exists an integer N 6 > N 5 such that if For the fourth iteration, we aim to show that V 1 ≥ N x 2 and V 2 ≥ N y 2 . From the first equation of system (4) we have A similar argument as above, we can obtain Hence, for any sufficiently small > 0, there exists an integer N 7 > N 6 such that if From the second equation of system (4) for n ≥ N 7 , we have Since m + cbN x 2 ≤ m + cbM x 1 ≤ 1, a similar argument as above, we can obtain Hence, for any sufficiently small > 0, there exists an integer N 8 > N 7 such that if Continuing the above process, we can finally obtain four sequences {M x n }, {M Clearly, we have that for any integer n > 0, Firstly, when n = 2, it is obvious Then Therefore Taking n → +∞ in (25), we obtain It follows from (26) that Subtracting the first equation of (27) from the second equation, we obtain Assume that x 1 = x 2 , (28) gives Substituting (29) into (27), we have where Therefore, x 1 and x 2 are two positive solutions of the following equation Since (20) holds, B 1 > 0 and B 3 < 0. Hence, (31) only has one positive solution, that is, x 1 = x 2 . And from (26), we have y 1 = y 2 .
Notice that the condition for the existence of E * is Hence, (19) holds implies the existence of interior equilibrium E * (x * , y * ), which satisfies (26). So we have x 1 = x 2 = x * and y 1 = y 2 = y * , than it follows U 1 = V 1 = x * and U 2 = V 2 = y * . Thus, we can obtain lim n→+∞ x(n) = x * , lim n→+∞ y(n) = y * , that is, the equilibrium E * (x * , y * ) is globally attractive. This completes the proof of Theorem 8.

Remark 1.
In this paper, we apply the comparison principle of difference equations to prove the globally asymptotic stability. However, in the process of inequality scaling, sometimes the scaling steps are too broad, leading to too narrow sufficient conditions. Therefore, a worthiness further study is whether the interior equilibrium E * is still globally stable when r 0 > 1 or hM It is an open problem which may be tested by using numerical simulation.

Global Stability of Prey Free Equilibrium
The previous section shows that populations of predators and prey can reach coexistence under certain conditions. However, a major feature of this paper is to consider that the predator has other food resource. In this section, we study the global stability of the prey free equilibrium E 1 (0, m h ), i.e., discuss whether predators population remain stable when the prey population becomes extinct.
We first state the following lemmas. Proof. Since (32) we get . Take the positive constants α and β such that Then we have αb − βh > 0.
And there exists δ > 0 such that For any p ∈ N, we can obtain Then from (33)-(35), we get Add both sides from 0 to n − 1, we have From Lemma 6, since For any sufficiently small > 0, there exists an integer N 1 > 0 such that if n ≥ N 1 , then Then from the second equation of system (4) for n ≥ N 1 , we have From Lemma 6, we obtain lim sup Hence, for any sufficiently small > 0, there exists an integer N 2 > N 1 such that if n ≥ N 2 , then y(n) ≤ exp(m + cbM 1 − 1) h + := M 2 . Then, the inequality (36) is equivalent to (40) shows that for sufficiently small > 0, x(n) → 0, when n → +∞. This ends the proof of Theorem 9.
Now we consider the following system whose any positive solution is y 1 (n) = m h . We can get the following theorem. Proof. Since (32) holds, we have lim n→+∞ x(n) = 0 according to Theorem 9. That is, for any sufficiently small > 0, there exists an integer N 3 > N 2 such that if n ≥ N 3 , then In order to prove lim n→+∞ [y(n) − y 1 (n)] = 0, we assume that Then the second equation of system (4) is equivalent to where θ(n) ∈ [0, 1]. Hence, y 1 (n) exp(θ(n)k(n)) is between y 1 (n) and y(n). To prove that lim n→+∞ [y(n) − y 1 (n)] = 0 is equivalent to proving lim n→+∞ k(n) = 0.
According to Theorems 9 and 10, the prey free equilibrium E 1 (0, m h ) is globally attractive.

Numerical Simulations
In this section, we show the feasibility of the main results in this paper. We show the impact of the fear effect on the discrete system visually through numerical simulations. In addition, a feature of this paper is that predators have other food resource, so we explore it in this section.

Example 1.
We show the feasibility of Theorem 7. In Figure 1a, the chosen parameters are: with initial value (0.8, 0.5), which satisfies the condition (1) of Theorem 7, i.e., r 0 > d + (dhkm + bhm + bkm 2 )/h 2 = 0.7425, T = 2.76 > 2 and 2T − 4 < S = 1.61 < T, with x * = 0.69, y * = 0.67. So that the interior equilibrium E * exists and is locally stable. In fact, we can see that E * is also globally stable in Figure 1a. Hence, regarding the open problem mentioned by Remark 1, we can verify that E * may be globally stable probably in the case of r 0 > 1.

Example 2.
In this example, we show the feasibility of Section 5. We set the parameter values as with initial value (0.8, 0.5), which satisfies the conditions (32) and (42), E 1 (0, 0.5) is globally stable in Figure 2. Moreover, set the parameter values as which satisfies the following conditions: i.e., the condition (2) of Theorem 5. Now we choose m as the bifurcation parameter to draw the plot. In Figure 3, the prey population tends to die out when m is large enough and the predator population becomes chaos as m increases. In addition, under (57) the condition (42) is not satisfied, so the chaos in Figure 3 can be used as a counter example of Section 5 to further prove the condition of Section 5, that is, E 1 is not globally stable if (58) holds.

Example 3.
To investigate the impact of the fear effect on system (4), we set the parameter values as with initial value (0.8, 0.5). We first choose the growth rate r 0 as the bifurcation parameter to draw the diagram without fear effect (k = 0). In Figure 4, the bifurcation plot shows that there is a single stable equilibrium when r 0 is less than 2.67. When r 0 reaches 3.2, the single stable equilibrium bifurcates into 2 stable points with a stable period-2 cycle. When r 0 reaches 3.32, the period-2 cycle bifurcates into a stable period-4 cycle. Then if r 0 > 3.32 the system (4) becomes chaotic dynamics without fear effect. In this case, we observe that as r 0 increases, the system (4) goes from stable to unstable.  Now we set r 0 = 4 to see if the original chaotic dynamics would become stable because of the presence and growth of k. In Figure 5, we observe that the system (4) changes from chaos to a stable period-4 cycle, then to a stable period-2 cycle, and finally to a stable equilibrium if we increase the value of k. Further, Figure 5 shows that when r 0 = 4 > 1 the system (4) may be still globally stable which just could illustrate the feasibility of Remark 1. In Figure 6, it is obvious that with the increase of k, the prey tends to become extinct, while the population of predators tends to a certain constant. It demonstrates the correctness of the global attractivity of prey free equilibrium in Section 5.

Example 4.
We still use the parameter settings of (58) to discuss about the impact of fear effect. Figure 7 shows that in domain I the interior equilibrium of system (4) is locally stable without fear effect. In this case, the populations of prey and predators eventually tend to positive constants under certain initial conditions. In domain II, as the fear level k gradually increases, E * changes from unstable state to asymptotically stable state, which illustrates that the cost of fear k can increase the stability of system.

Conclusions
In this paper, we have studied a discrete-time predator-prey system incorporating fear effect of the prey with the predator has other food resource. System (4) is proposed based on the continuous system (3), which was proposed by Zhu et al. [32]. It is shown that the system (4) has a trivial equilibrium for all positive parameters which is always unstable. This property is similar to the conclusion of Zhu et al. [32]. Meanwhile, there exists a prey free equilibrium for all positive parameters and it may be locally stable for some values of parameters while it becomes unstable for other parametric values. System (4) has a predator free equilibrium and a positive equilibrium for some values of parameters. We get the existence condition of positive equilibrium is r 0 > [d + (dhkm + bhm + bkm 2 )/h 2 ] as same as the result of Zhu et al. Moreover, the rigorous mathematical proof of the global attractivity of the positive equilibrium is given by using iteration scheme and the comparison principle of difference equations. Compared with the results of continuous system (3), we obtain more strict sufficient conditions. However, since we scale the equation when using the comparison principle, it may leads to too strict for the global stability condition of positive equilibrium. In this regard, we propose Remark 1 and verify the conjecture of Remark 1 in the numerical simulation.
A major feature of this paper is to consider that the predator has other food resource. So we specifically study the stability of the prey free equilibrium. In [32], Zhu et al. studied locally and globally asymptotically stable of the prey free equilibrium of continuous system (3). Their result (see Theorem 1) indicates that E 1 (0, m h ) remains stable if r 0 < [d + (dhkm + bhm + bkm 2 )/h 2 ] holds. However, in this paper, we obtain Theorems 5, 9 and 10 which shows that the stability of E 1 is related not only to the value of r 0 , but also to the value of m, i.e., the intrinsic grow rate of the predator species. Analytical results and numerical simulation (Figure 3) show that the prey and the predator may go extinct at the same time if the intrinsic grow rate of the predator is not large enough. With the increase of m, the prey population first appears chaos and then tends to die out while the predator population becomes chaos. Although r 0 < [d + (dhkm + bhm + bkm 2 )/h 2 ] holds, E 1 is not globally stable as the intrinsic grow rate of the predator increasing. This is the difference between this paper and Zhu et al.'s. Thus, we can also confirm what we stated in the first section: although the dynamic behaviors of discrete and continuous systems are similar in general, there are some differences that discrete systems often have more complex dynamic behavior than continuous.
On the other hand, we consider the effect of the value of fear parameter k on the dynamic behavior of the discrete system (4). In Section 6, the influence of fear effect is visually demonstrated. From Figures 4-7, it is very intuitive to see that the fear effect can turn an unstable system into a stable one. In other words, the fear effect enhances the stability of the discrete system (4).