Multiple Periodicity in a Predator–Prey Model with Prey Refuge

: We consider a delayed prey–predator model incorporating a refuge with a non-monotone functional response. It is supposed that prey can live in the predatory region and prey refuge, respectively. Based on Mawhin’s coincidence degree and nontrivial estimation techniques for a priori bounds of unknown solutions to the operator equation Lv = λ Nv , we prove the existence of multiple periodic solutions. Finally, an example demonstrates the feasibility of our main results.

To this day, the Lotka-Volterra type system is one of the important themes in mathematical biology. Many scholars have made contributions to it (see, e.g., [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]). In the interaction between prey and predator, there always exists a phenomenon of prey refuge. In general, the entire prey population lives in two areas: the predatory region and the prey refuge. From a biological view, the prey refuge can exist without predators; therefore, it can help improve the population density of the prey. In addition, a prey refuge is an effective strategy to reduce predation in the evolution of prey population. For this reason, it was proposed by Gause et al. [22,23]. Moreover, Magalhães [24] considered the refuge effect on the dynamics of thrips prey and mite predators. Ghosh et al. [25] studied the influence on a predator-prey system of adding extra food for predators and incorporating a prey refuge. Xie [26] investigated a prey-predator model incorporating fractional-order factors and a prey refuge. When in a high-prey refuge ecological system, Sahoo et al. [27,28] observed that the possibility of predator extinction could be eliminated by providing additional food to the predator population. Motivated by these works, Jana et al. [29] considered the following prey-predator model with prey refuge: where u is the density of the prey in the refuge and v is the density the prey in the predatory region; w denotes the density of the predator in the predatory region; the intrinsic growth rate for u and v is denoted by r 1 and r 2 , respectively; the prey migrating in the refuge to the predatory region is given by σ 1 and migrating from the predatory region to the refuge is denoted by σ 2 ; k 1 and k 2 are the environment carrying capacity for u and v, respectively;d and γ are the natural death rate and the density dependent mortality rate of predator, respectively; the rate of the predator consuming prey is denoted by β (suppose that 0 < β ≤ α); τ is the delay; and a and α are two parameters in Holling type II functional response.
On the other hand, functional response is also an important factor affecting the predator-prey model. In [30], we proved that there exists at least one positive periodic solution for Jana's model with a Holling type II functional response. However, there exists a functional response unlike the Holling functional responses. Some experiments have indicated that it may occur at the microbial level: when nutrient concentrations reach a high level, they may inhibit specific growth rates; see [31]. Hence, the non-monotone functional response was considered and used to model the inhibitory effect at high concentrations [32][33][34]. In addition, in real-world applications, some researchers believe that predators living in the predatory region are classified by two fixed ages: one is mature predators, and the other is immature predators-the immature predator have no ability to attack prey; see [35][36][37][38][39].
Motivated by the works of Jana et al. [29] and Chen [34], in the present paper, we establish the following delayed stage-structured prey-predator model with a prey refuge and non-monotone functional response: where z 1 (t) and z 2 (t) are the density of the immature predator and mature predator at time t, respectively.
, and d 2 (t) are all continuously positive periodic solutions with period ω. k(t) and d(t) are the death rate of the predator. Moreover, the non-monotone functional response is αy y 2 /m+y+a . The term indicates the number of immature predators that were born at time (t − τ) that still survive at time t and then become mature predators. Moreover, the term t t−τ k(s)ds denotes the stage-structured degree of the immature predator; one can refer to Liu et al. (see [11] pp. 670-671).
In terms of the number of creatures, the initial conditions are associated with It is well known that the global existence of periodic solutions is a very basic and important problem in the study of periodic population dynamics. Because periodic environments such as the seasonable effect are important factors, the existence of periodic solutions plays a similar role to a global equilibrium in an autonomous model. Thus, the aim of the present paper is to find some suitable conditions of the existence of positive periodic solutions for system (2). Based on Mawhin's coincidence degree theory and other nontrivial techniques, we prove the existence of multiple positive periodic solutions for system (2) in Section 2. In Section 3, we propose some examples to demonstrate the feasibility of our main results.

The Existence of Multiple Positive Periodic Solutions
Firstly, we separate the third equation of system (2) from the whole system and obtain the following subsystem: The initial values for system (3) are To obtain the multiple positive periodic solutions of system (3), we summarize the following lemmas.

Lemma 1.
Let Ω ∈ V be an open bounded set. Assume that L is a Fredholm operator of index zero and N is L-compact onΩ. Furthermore, if the following conditions are fulfilled then the operator equation Lv = Nv has at least one solution inΩ ∩ DomL.
Note that operator L is said to be a Fredholm operator of index zero if dim ker L = codim L < ∞ and L is closed in V. If L is a Fredholm mapping of index zero, then there exist continuous projectors P : U → U and Q : V → V such that P = ker L and L = ker Q = (I − Q), where U, V are Banach spaces. For further symbolic meaning of the concepts in Lemma 2.1, one can refer to [40][41][42] for details.

Lemma 2 ([32,41]).
Assume that x(t) is an ω-periodic function that is continuously differentiable. Then, there exists a t ∈ [0, ω] such that For the sake of convenience, we use notations as follows.
As mentioned in the last paragraph of the introduction, periodic solutions are of great importance. It is reasonable to seek conditions (that is, (H 1 ), (H 2 )) under which the resulting periodic system would have positive periodic solutions. From a biological viewpoint, the assumption (H 1 ) implies that the rate of predator feeding on prey is affected by the death rate of predators, the time delay, and the migration of prey, while the assumption (H 2 ) implies that the intrinsic growth rate of prey in a predator region is influenced by prey migration, environmental carrying capacity, and predator predation. Now, we are in a position to state our main theorem.
Theorem 1. If (H 1 ) and (H 2 ) hold, then system (2) has at least two positive periodic solutions.
Proof. We prove this theorem into two steps.
Step 1 We claim that there are at least two periodic solutions of subsystem (3). In fact, by the variables transformation then V, U are both Banach Spaces with the norm || · || as follows: it is easy to see that and are all ω-periodic functions. Indeed, Clearly, Θ 2 (v, t), Θ 3 (v, t) are also periodic functions in a similar way. Set It follows form the above definitions that ker L = {v ∈ V|v = C 0 , C 0 ∈ R 3 } and Obviously, QN and K p (I − Q)N are continuous. It follows from the Arzela-Ascoli theorem [34,40] that N is L-compact onΩ with any open-bounded set Ω ⊂ V.
The next work is to find an appropriate open-bounded subset Ω for the application of the continuation theorem. Corresponding to Lv = λNv for some λ ∈ (0, 1), we obtain

is a solution of Equation (4). Integrating Equation (4) over the interval [0, ω] leads to
From the first equation of (4) and (5), we have Similarly, from the second equation of (4), (5) and from the third equation of (4), (5), the following inequalities hold: Combining Lemma 2 with these inequalities, we next construct the upper and lower bounds of subsystem (3).

This can be rewritten as
In view of (H 1 ), we obtain Let Equation (13) equal 0; we see that Hence, we take max From the first equation of (5), we writē It follows from (6), (14), and Lemma 2.2 that In a similar way, we obtain the following inequality from the first equation of (5), From (6), (15) and Lemma 2.2, we have Therefore, we take max t∈[0,ω] |v 1 (t)| < max{B 11 , B 12 } := B 1 .

Step 2
We now claim that the third equation of (2) has two ω-periodic solutions associated with the obtained solutions. Let h(t) = −k(t), Then, the third equation of (2) is given by In fact, we see that Similarly, h + (t + ω) = h + (t), g + (t + ω) = g + (t).

Conclusions
In this paper, a delayed prey-predator model incorporating a refuge with a nonmonotone functional response is considered. It is assumed that prey can live in the predatory region and the prey refuge, respectively. From the biological point, a prey refuge can help to improve the population density of the prey, and it is an effective strategy to reduce predation in the evolution of prey population. Based on the method of Mawhin's coincidence degree theory and non-trivial estimation techniques for a priori bounds of unknown solutions to the operator equation Lv = λNv, we obtain some interesting and novel sufficient conditions for the existence of multiple periodic solutions of the prey-predator model. However, the limitation of this method is that we cannot determine the specific number of periodic solutions. However, predictably, for different biological models, we can give a lower bound on the number of periodic solutions. In addition, we believe that, in this paper, the occurrence of two periodic solutions is influenced by non-monotonic functional response. These two periodic solutions are generated by the system for parameters in different parameter ranges, but the periods are the same. Namely, in the bounded open subset Ω 1 , system (2) presentws a positive periodic solution; in another bounded open subset Ω 2 , system (2) also has a positive periodic solution with the same period.
Author Contributions: Methodology, W.L. and Y.X.; software and figure, W.L.; validation, W.L. and Y.X.; formal analysis, W.L. and Y.X.; investigation, W.L. and Y.X.; resources, W.L. and Y.X.; writingoriginal draft preparation, W.L. and Y.X.; writing-review and editing, W.L. and Y.X.; supervision, Y.X. All authors have read and agreed to the published version of the manuscript.