The Study of a Predator-Prey Model with Fear Effect Based on State-Dependent Harvesting Strategy
In presence of predator population, the prey population may significantly change their behavior. Fear for predator population enhances the survival probability of prey population, and it can greatly reduce the reproduction of prey population. In this study, we propose a predator-prey fishery model introducing the cost of fear into prey reproduction with Holling type-II functional response and prey-dependent harvesting and investigate the global dynamics of the proposed model. For the system without harvest, it is shown that the level of fear may alter the stability of the positive equilibrium, and an expression of fear critical level is characterized. For the harvest system, the existence of the semitrivial order-1 periodic solution and positive order- () periodic solution is discussed by the construction of a Poincaré map on the phase set, and the threshold conditions are given, which can not only transform state-dependent harvesting into a cycle one but also provide a possibility to determine the harvest frequency. In addition, to ensure a certain robustness of the adopted harvest policy, the threshold condition for the stability of the order- periodic solution is given. Meanwhile, to achieve a good economic profit, an optimization problem is formulated and the optimum harvest level is obtained. Mathematical findings have been validated in numerical simulation by MATLAB. Different effects of different harvest levels and different fear levels have been demonstrated by depicting figures in numerical simulation using MATLAB.
Prey-predator interaction is a crucial topic in theoretical ecology and evolutionary biology. The history of the study about the prey-predator interactions dates back long. The pioneering work to describe the prey-predator interactions in mathematics belongs to the Lotka-Volterra model [1, 2]. Subsequently the model was improved by adding logistic growth term for the prey and variety of population-dependent response functions [3–15]. A prototype model that captures the prey-predator interaction takes the formwhere and represent the densities of prey and predator population, respectively, , , () and represent the birth rate, natural death rate, and density-dependent decay rate due to the intraspecies competition, respectively, represents the functional response, is the efficiency of conversion, is natural mortality of predator, and is a monotonically increasing function.
Due to prey-predator interactions, predators always have an impact, direct, indirect, or both, on prey population. In model (1), the term models the direct impact of predator on prey by catching and killing behavior. Meanwhile fear of predation risk can be regarded as the indirect impact of predator on prey, and some theoretical ecologists and biologists have realised that a prey-predator model should involve not only direct killing but also the fear [16, 17]. The fieldwork of Zanette et al.  on song sparrows observed the impact of fear and found a reduction in reproduction by 40% in the number of offspring due to the fear of predation. Based on this phenomenon, Wang et al.  incorporated a predator-dependent fear factor into the birth rate of prey in model (1) (i.e., replace by ) with linear and Holling type-II functional response to explore the effect of fear on population dynamics. The results show that high level of fear could stabilize the system. Das and Samanta  investigated the impact of fear in exponential form on a stochastic prey-predator system when the predator is provided additional food. Sahoo and Samanta  investigated a two prey-one predator model by including the cost of fear into prey reproduction and switching mechanism in predation. Das et al.  developed and explored a predator-prey model incorporating the cost of perceived fear into the birth and death rates of prey species with Holling type-II functional response. Sarkar and Khajanchi  and Kumar and Kumari  incorporated a form of fear factor into the birth rate of prey by assuming a nonzero minimum cost of fear. The impact of fear has also been investigated on prey-predator systems with prey refuge [25–27], Allee effect , hunting cooperation , and additional food resource for predator [20, 29].
The study of resource management including fisheries, forestry, and wildlife management has great importance. It is necessary to harvest the population but harvesting should be regulated in such a way that ecological sustainability as well as conservation of the species can be implemented in a long run. Besides, it is always hoped that the sustained ability can be achieved at a high level of productivity and good economic profit. In the past decade, scholars considered different kinds of harvest on the dynamics of the predator-prey system such as continuous harvesting [30–33] and intermittent harvesting [34–37]. Compared to fixed time harvest strategy, the state-dependent harvesting strategy takes the existing resources of species into full consideration and can maintain the sustainability of species in certain level. State-dependent harvested system can be described by the impulsive semidynamical system [38–45]. Recently, Lai et al.  proposed and studied a Lotka-Volterra predator-prey system incorporating both continuous harvesting and fear effect; that is,where is the level of fear, is the fishing effort used to harvest, is the catchability coefficient, and and are constants. The harvest in (2) is continuous and in Michaelis-Menten type. However, in reality, the harvest of species should consider the aspect of ecological sustainability as well as conservation. Thus, in most cases, species are caught intermittently, not continuously.
To the best of our knowledge, to this day, still no scholars investigated the dynamic behavior of the predator-prey system incorporating both fear effect and intermittent harvesting, which motivated us to study a predator-prey model incorporating fear effect based on state-dependent harvesting strategy. The aim of this study is to check the influence of fear level on the stability of the positive steady state of the system without harvest. Meanwhile, for the harvest system, it mainly discusses the existence of the order- () periodic solution, since it provides a possibility to transform the state-dependent harvesting into a cycle one. Meanwhile, in order to make a maximum economic profit in the harvest process, the optimal control problem is discussed. The organization of this study is as follows. In the next section, we introduce the mathematical model for predator-prey system with fear effect based on state-dependent harvesting strategy. In the same section, we present some preliminaries used in the discussion of the system dynamics. Section 3 is dedicated to the existence and stability of semitrivial order-1 and positive order-1 periodic solution. We also study the existence of order-2 and order-3 periodic solution. In Section 4, we demonstrate different effects of different harvest levels and different fear levels by depicting figures in numerical simulation using MATLAB. The paper concludes in Section 5, in which we briefly summarize the biological indications of our analytical findings.
2. Model Formulation and Preliminaries
2.1. Model Formulation
In presence of predator population, the prey population may significantly change their behavior. Fear for predator population enhances the survival probability of prey population, and it can greatly reduce the reproduction of prey population . In this study, we consider a predator-prey model introducing the cost of fear into prey reproduction with Holling type-II functional response and a saturation function in equation (1); that is,Where the variables, model parameters, and their units/dimensions are given in Table 1. To achieve the commercial purpose of the fishery, it is necessary to harvest the population in such a way that ecological sustainability as well as conservation of the species can be implemented in a long run. The harvest can be continuous or intermittent. In this work, a state-dependent harvest strategy is considered. Let be the harvest level of prey population; that is, when the density of prey population reaches level , the harvest is implemented, resulting in a portion of prey and predator being caught. Let denote the harvest effort, which is dependent on the harvest level , and let and be the catchability coefficients of prey and predator populations. In addition, to avoid the extinction of predator, it is necessary to release a quantity of predator pups, denoted by , which is also dependent on level . Based on this consideration, the model with state-dependent harvesting takes the following form:
Denote . Then is the carrying capacity of prey population in absence of predator. System (4) is considered in the domain for ecological practices. The purpose of this paper is to analyze the dynamics of system (4). Besides, it is always hoped that the harvest can be achieved at a good economic profit, and this requires determining an optimal harvest level . Next, some preliminaries are listed for the analysis of the harvest model (4).
Let us consider a general planar system:where , and describes the states at which the harvest is implemented; and describe the effects of the harvest strategy. and are arbitrarily derivative with respect to ; , , and are linearly dependent on and ; that is, , , , , , and are constant.
The dynamic system constituted by the solution mapping defined by system (5) is called an impulsive semicontinuous dynamic system, denoted as , where : , , and
Let be the solution of system (5) with initial value . Denote , also denoted as in short. Denote , where with .
Definition 1 (priodic solution [47–49]). The solutionof system (5) is said to be periodic if there exists positive integer such that. Denote; then orbitis said to be an order-periodic orbit of system (5).
Definition 2 (orbitally stable [47–49]). An orbitis said to be orbitally stable if, for any, there is a neighborhoodofso that, for allin, there is a reparameterization of time (a smooth, monotonic function) such that.
Definition 3 (asymptotic orbital stability [47–49]). is said to be asymptotically orbitally stable if it is orbitally stable and additionallymay be chosen so that, for all, there exists a constantsuch that.
Definition 4 (Poincaré map). Let. Define the Poincaré map:as follows:.
Remark 1. If there exists a pointandsuch thatand(), that is,is a fixed point of, then system (4) admits an order-periodic solution.
Lemma 1 (analogue of Poincaré criterion [47–49]). The order--periodic solutionof system (5) is orbitally asymptotically stable and enjoying the property of the asymptotic phase if the multipliersatisfies the condition, wherewithand ,, and,,,are calculated at the point.
3. Main Works
Then is called the critical value of the conversion; that is, when , the conversion is not enough to maintain the survival of predator and the predator population will go to extinction. Thus, in this work, it is reasonable to assume that .
For system (3), the following result holds.
Theorem 1. There are three equilibria for system (3) when: two boundary saddlesandand one positive equilibrium. Moreover, one of the two following cases holds:(i)is a stable focus or node in case of(ii)is unstable in case of, and a unique stable limit cycle exists, denoted by
Proof. The Jacobian matrix of model (3) at the equilibrium isIt is obvious that and are saddles. At the equilibrium , the characteristic equation . If holds, then . By equation (6), if and only if . Thus, the positive equilibrium is locally asymptotically stable in case of and unstable in case of . In this case, there exists a unique stable limit cycle for system (3).
3.1. Semitrivial Order-1 Periodic Solution for
When , there is for with . In this case, system (3) is reduced to the following system:
Setting , the solution of equation with is
Then there is and by impulse effect. Thus, the following result holds.
Theorem 2. System (4) withhas a semitrivial order-1 periodic solution for:which is orbitally asymptotically stable when , where
Proof. To discuss the stability of , let us consider a small disturbance . The trajectory starting from is denoted by . This disturbed trajectory first intersects the harvest set at point when , and then it jumps to point . Thus, there isLet and . Then , and . Setting , for , the variables and can be expressed by the relationwhere is the fundamental solution satisfying the variation equation.According to the first-order Taylor expansion on , there is , whereBy impulse effect, there is . Thus, if inequality (10) holds, there is . By the arbitrary of , it concludes that the order-1 semitrivial periodic solution is orbitally asymptotically stable.
Corollary 1. The semitrivial order-1 periodic solutionis orbitally asymptotically stable if one of the two following cases is satisfied: (i)and (ii)and.
3.2. Positive Order-K Periodic Solution for
Since the harvest may cause the extinction of predator when , in order to keep the predator species from going extinct, it is necessary to reduce the harvest strength and release a certain quantity of predator pups.
For , define
Let and denote the intersection point between and the phase set and the harvest set , respectively; denotes the intersection point between and the phase set ; in general . For a point on with , if the trajectory of system (4) starting from intersects the harvest set , then it defines a function relationship between and for , denoted by , which satisfies
By equation (22), the function can be expressed as follows:
Define . When , the trajectory of system (4) starting from will intersect the harvest set , and denote the intersection point by ; that is, for some . Define .
3.2.1. Existence of Order-1 Periodic Solution
By Theorem 1, the dynamic behavior of system (3) varies with the model parameter . Thus, the discussions will be divided according to parameter and harvest level . Case I: Case I-1:
Since on , map in Definition 4 is only a function of . Next, the Poincaré map will be characterized and its main property will be analyzed.
For , by Definition 4, there is . Meanwhile, for , there exists a unique and such that . Then . To sum up, there is
Property 1. For system (4), when , the Poincaré map defined by equation (24) has the following properties:(i)is continuous on. Moreover,is increasing on and decreasing on(ii)is continuously differentiable on , and is concave on(iii)There exists a horizontal asymptote ; that is, when Define . The following result holds.
Theorem 3. There exists a unique positive order-1 periodic solution for system (4) whenand.
Proof. By Remark 1, the existence of order-1 periodic solution is equivalent to the existence of a point such that is a fixed point of . By Property 1 (i), is continuous on . Since and as , by the intermediary property of continuous function, there exists at least one such that ; that is, . Thus, the trajectory of system (4) starting from forms an order-1 periodic orbit.
Next, the location and uniqueness of the order-1 periodic orbit will be analyzed. By Property 1 (i), achieves its maximum at . It is obvious that .
If , then .
If , then , which means that , as shown in Figure 1(a). Since is concave on , is unique.
If , there is ; that is, . Since is decreasing on , ; that is, . Besides, define and . Denote . Then there is and is unique, as shown in Figure 1(b)).
Case I-2: : in case of , the trajectory of system (4) starting from will intersect the harvest set . When , the trajectory starting from point does not intersect the harvest set . Denote and . Then the domain of is . Define and .
Theorem 4. There exists a positive order-1 periodic solution for system (4) when (i)andor (ii)and.
Proof. When , similar to the proof of Theorem 3, system (4) admits an order-1 periodic solution. For , if , then, for , there is . Combining with , it can be concluded that system (4) admits an order-1 periodic solution. For , there is and ; thus, there exists such that ; that is, system (4) admits an order-1 periodic solution. □
Case II: : in this case, the trajectory of system (4) starting from will intersect the harvest set .
Theorem 5. There exists a positive order-1 periodic solution for system (4) when (i)andor (ii)and. Moreover, the order-1 periodic solution is unique when.
3.2.2. Stability of the Order-1 Periodic Solution
Let be an order-1 periodic solution of system (4). Denote , , , and . Define
Theorem 6. The order-1-periodic solutionis orbitally asymptotically stable if
Theorem 7. For, if, the order-1 periodic solution for system (4) is globally orbitally asymptotically stable.
Proof. By Theorem 3, system (4) admits a unique order-1 periodic solution when . If , there exists a unique such that . Thus, for any , a sequence is obtained under ; that is, . If , then is a monotonically increasing sequence with , so the limit is . Similarly, if , then is a monotonically bounded decreasing sequence, and the limit is . If , then ; thus is a monotonically bounded sequence with limit . To sum up, by the arbitrariness of , the order-1 periodic solution is globally attractive and so is globally orbitally asymptotically stable.
3.2.3. Existence of Order- () Periodic Solution
For , by Theorem 3, if , the order-1 periodic solution is orbitally asymptotically stable and globally attractive, which means that system (4) does not admit order- () periodic solution. For , there exists unique such that . Let such that . Then . Meanwhile, let and such that .
Theorem 8. Forand, if (i)or (ii)andholds, system (4) admits an order-2 periodic solution.
Proof. Obviously, . It can be easily checked that is increasing on and , and is decreasing on and .(i). In this case, there is ; that is, ; then . Besides, there is . Thus, there exist and such that and . Moreover, there are and .(ii). In this case, there is ; that is, . For any , there is . Next, it mainly discusses the property of on . Let . Then , , and . Then, under , a sequence is obtained, whereDenote and . It is obvious that . Since , then . Moreover, there is and . It can be concluded that , that is, the order-2 periodic solution is orbitally asymptotically stable and globally attractive.
Theorem 9. Forand, system (4) admits an order-3 periodic solution if and only if. Moreover, there is at least one order-3 periodic solution when, and there are at least two order-3 periodic solutions when.
Proof. “Necessity.” Proof by contradiction. Assume that .
Since , if , system (4) admits a stable order-1 periodic solution or a stable order-2 periodic solution. Moreover, there is . If , that is, , there exist and such that . It can be easily checked that is increasing on , , and , and is decreasing on , , and . If , then does not exist, is increasing on and , and is decreasing on , , and . In any case, since , , and , it can be concluded that if and only if ; that is, the order-3 periodic solution does not exist.
If , system (4) simultaneously admits an order-1 periodic solution and order-2 periodic solution. Moreover, there is . If , that is, , there exist , , , and such that . It can be easily checked that is increasing on , , , and , and is decreasing on , , , and . If , then does not exist, is increasing on , , and , and is decreasing on , , , and . Since and , it can be concluded that if and only if ; that is, the order-3 periodic solution does not exist.
“Sufficiency.” If , then there is and . If , then there exists an order-3 periodic solution since . If , then there exists at least one such that and ; that is, system (4) admits an order-3 periodic solution.
Next, the number of order-3 periodic solutions will be discussed:(i)When , there are , , and . Moreover, there are and ; that is, system (4) admits at least one order-3 periodic solution.(ii)When , there are , and , , , and