Abstract
In this paper, we propose a stage-structured predator-prey model with migrations among patches in an -patch environment. The net reproduction number for each patch in isolation is obtained along with the net reproduction number of the system of patches, . Inequalities describing the relationship among these numbers are also given. Furthermore, threshold dynamics determined by is established: the predator dies out if while the predator persists if . Focusing on the case with two patches, we obtain that the dispersal decreases the net reproduction number . By numerical simulations, we find that the dispersal may be a good thing or a bad thing because the dispersal could make the predator population thrive or extinct, and hence we might seek steady state in the ecological environment by controlling parameters related to the prey and the predator.
1. Introduction
The effect of dispersal of organisms on population dynamics is one of the central topics in ecology and evolutionary biology [1–5]. Each species has a characteristic pattern of variation in abundance over space. Understanding the factors responsible for manifest diversity in distributional patterns is an important prerequisite for studying survival and extinction of species. For example, many types of birds and mammals migrate from cold regions to warm regions in search of a better habitat or a breeding site [6], and for ecological communities of insects, dispersal of a predator is usually driven by its nonrandom foraging behavior which can often respond to prey-contact stimuli [7, 8].
During the past couple of decades, due to the prevalence of the predator-prey relationship in the nature, predator-prey models with diffusion in patchy environment have attracted significant attention from ecologists, biologists, and biomathematicians. Levin [9] proposed two-species competition and prey-predator models with population dispersal among patches. Takeuchi [10] and Kuang and Takeuchi [11] found that stabilizing and destabilizing effects could be induced by prey dispersal. Gramlich et al. [12] considered a 2-patch 2-species system and found that dispersal between different habitats influences the dynamics and stability of populations considerably. Moreover, these effects depend on the local interactions of a population with others. Kang et al. [8] formulated a two-patch Rosenzweig–MacArthur prey-predator model with mobility only in the predator, and they found that dispersal may stabilize or destabilize the coupled system; dispersal may generate multiple interior equilibria or may destroy interior equilibria. In a few patch models, the common findings for most studies on interactions of prey and predator are that dispersal makes the dynamics very complicated. This means that dispersals of prey and predator play an important role in regulating, stabilizing, or destabilizing population dynamics of both prey and predator. In an -patch environment, due to the intricacies that arise from dispersal, not much has been done yet. Qiu and Mitsui [13] considered a predator-prey system with diffusion and time delay in an n-patch environment, where prey disperses between patches of a heterogeneous environment with barriers between patches where a predator cannot cross. Al-Darabsah et al. [14] proposed a prey-predator model in multiple patches through the stage-structured maturation time delay with migrations among patches. However, they only discussed the existence of equilibrium points and the uniform persistence for the special case of two patches.
To our best acknowledgment, little attention has been paid to the relation between the long-term dynamics and dispersal rate of the multipatch predator-prey model. In two recent references [15, 16], net reproduction number was introduced into the nondiffusion predator-prey models; it was shown that the net reproduction number determines its threshold dynamics; here has the similar definition to the basic reproduction number of general compartmental epidemic model, as discussed in [17]. Since there are many excellent results on investigating the effects of mobility rate on basic reproduction number and the threshold dynamics of multipatch epidemic model, for reference, we refer to Hsieh et al. [18], Wang et al. [19], and Wang et al. [20]. This motivates us to study an -patch predator-prey model with prey and predator diffusing simultaneously.
In this paper, we focus on a model with prey and predator dispersal in an -patch environment. In addition, the predator is structured as juvenile and adult. We assume that juvenile predators may diffuse as they can follow adult one. We also need to pay attention to acquisition probability of the juvenile predator, which can be zero and less than unity. Moreover, the dispersal among patches may occur only in a single species. The main focus of our study of such prey-predator interactions in an -patch heterogeneous environment is to explore the following ecological questions:(1)What is the net reproduction number ? And whether there is a limit on ?(2)In a two-patch model, what is the relationship between the constant dispersal rates of prey and predator and ?(3)How does dispersal of prey and predator affect the extinction and persistence of predator in all patches?(4)How may dispersal promote the coexistence of prey and predator when predator goes extinct in a single patch?
The paper is organized as follows. In the next section, we propose the model and show the positiveness and ultimate boundedness of solution under some conditions. In Section 3, we derive a general formula for the net reproduction number and give a bound for , show the local asymptotical stability of the predator-free equilibrium when , and also obtain the global attractivity of the predator-free equilibrium under some conditions. Moreover, we consider the special case with the linear predation function in a two-patch environment. In Section 4, we prove the uniform persistence when , and hence the system has at least one positive equilibrium. Numerical simulations are provided in Section 5 to demonstrate our results. The paper concludes with a brief discussion.
2. The Model
In this section, we formulate a predator-prey model in patches by taking into consideration diffusion among patches and a stage structure in the predator. In each patch , there are prey individuals, juvenile predator individuals, and adult predator individuals, denoted by , , and , respectively.
For patch , let be the intrinsic growth rate in the absence of predation and be the carrying capacity of the prey; , , and denote the death rates of prey, juvenile predators, and adult predators, respectively; we always assume ; stands for the acquisition probability of juvenile predators and ; we always assume to be the probability of the captured prey being converted into juvenile predators; is the progression rate of juvenile predators. All parameters are assumed to be positive except that can be zero.
We assume that the predation function satisfies the following basic assumptions for .(1), .(2), is a continuous function about and , .(3), .
The following three types of predation functions in [21] satisfy the above assumptions.(1), .(2), .(3), .
We assume that there is no capture and captured behavior in the diffusion process, and for are the constant dispersal rates from patch to patch for of prey, juvenile predators, and adult predators, respectively, with . It is assumed that the matrices , for are irreducible. The flowchart of diffusion is shown in Figure 1.
Based on the above assumptions, our model is as follows:
Similar to [19], for a vector , we use to denote the diagonal matrix, whose diagonal elements are the components of . We use the ordering in generated by the cone , that is, if , if and , and finally means for any index .
Denote the vector
Moreover, we assume that each component of is nonnegative with the following initial conditions:
Then, it follows from the standard existence and uniqueness theorem for ordinary differential equations that there is a unique solution to system (1).
Theorem 1. Consider system (1) with nonnegative initial conditions (4). Then, for each and , , , and are all positive and ultimately bounded.
Proof of Theorem 1. Firstly, we prove that for , for . For convenience, we rewrite the first equation of (1) aswhereLet and . Since , we know that . If , then from (5) with , we easily see that for . If , then we are done. Now suppose that . Since is irreducible, there exists such that for some . Then, for . This and (5) with imply that for . Denote and . If , then we are done. Otherwise, continuing this way, after a finite number of steps, we have for and .
Next, we prove that for , and for . Since , there exists such that . First, we assume that . By notingwhich is similar to (5), we can use similar arguments as those of showing for and to obtain for and . Then, fromwe see that for and . Now, assume that , which implies that . Then, from the second equation of (1), we get for . This, combined with the third equation of (1), yields for . Then as before, we can show that and for and .
Finally, we show the boundedness of solutions. On the one hand, let . Then, by the Cauchy–Schwartz inequality, we haveIt follows that . On the other hand, let . Then,It is easy to see from this and thatThis completes the proof.
To show the existence of predator-free equilibria (PFE), we let in equation (1) to getIn order to obtain a positive equilibrium for system (12), we assume thatwhere represents the stability modulus of an matrix and is defined by is an eigenvalue of for an matrix . From [19, 20, 22], we have the following two lemmas.
Lemma 1. Under assumption (13), system (12) admits a unique equilibrium , which is positive and globally asymptotically stable in .
Lemma 2. Under assumption (13), system (1) admits a unique PFE , where is the unique equilibrium of (12) and .
3. Threshold Dynamics
3.1. The Net Reproduction Number
In this section, we first give definition of the net reproduction number for the general predator-prey dispersal model (1), which is similar to the basic reproduction number for epidemic system; for reference, we refer to [17, 23].
Let be the PFE of (1). From [17], we havewhere
Note that and are both irreducible nonsingular -matrices with positive column sums and hence and . Following the procedure introduced in [17], the net reproduction number of equation (1) is given by , where represents the spectral radius of the matrix. Let . Then, by [17], there hold two equivalences:
Aswe have
In equation (19), accounts for the contribution of juvenile predators to the adult predators, while accounts for the contribution by adult predators themselves. Diffusion rates of preys influence the predator population capture rate, and diffusion rates of predators affect their consumption.
The net reproduction number gives an important threshold of the -patch diffusion predator-prey system.
Theorem 2. Consider model (1). Suppose (13) holds. If , then the PFE is locally asymptotically stable, and if , then the PFE is unstable. Moreover, if , then the PFE is globally attractive for solutions with initial conditions (2).
Proof of Theorem 2. It follows from ([10], Theorem 2) that the PFE is locally asymptotically stable if and unstable if .
Now, we prove the global attractivity of the PFE as follows.
Let be an arbitrary solution of (1). We first show that .
Since , there exists an such that , whereNote that satisfieswhere . By Lemma 1 and Comparison Theorem [24, 25], we have . It follows that there exists such that for . Then, for , satisfiesSince , we obtain by applying Comparison Theorem [24, 25].
Next, we prove that . It suffices to show as . Since is continuous and is bounded, , there exists such that for . By equation (13), there exists such that for . For any , there exists such that for . Then,for and . By Lemma 1 and Comparison Theorem [24, 25], we know that , where is the positive equilibrium of (12) with being replaced by , . It is easy to see that . Therefore, we have as required.
To summarize, we have shown that . This completes the proof.
Next, we consider the effect of dispersal on and provide lower and upper bounds of . In the rest of this paper, we let for . When there is no diffusion of prey, that is, , we have with for and the net reproduction number in patch in isolation is given byWe define a modified reproduction number in patch :Clearly, . Then, we can obtain the following result on the bounds for the net reproduction number for system (1), in terms of the numbers defined in equations (24) and (25) for each patch.
Theorem 3. For system (1), if , thenMoreover, if , , , , , , , and for , then
Proof of Theorem 3. From [18, 26], let denote the matrix obtained by deleting the first row and the first column of (); and denote and , respectively. Let , where and . It follows that and for . By [26], we haveHere,By virtue of Fisher’s inequality [27],Then,Similarly,Thus, we haveAgain, by Fisher’s inequality,Then, from these conclusions, we knowSimilarly, it can be shown thatand (26) is proved.
If we let , , , , , , , and for , then for . Without loss of generality, assume that . From the fact that the sum of each column of the matrix is equal to and the sum of each column of the matrix is , we see that and for . Thus, for the matrix , the sum of column isSimilarly, we can prove . From [26], we obtain the following inequalities:Combining this with (26), we get (27). This completes the proof.
3.2. The Model with Two Patches
In this section, we consider a special case of equation (1) with and assume that , for , namely,
The PFE of equation (39) is , where satisfies
Let
Note that
Recall that for . Hence, we havewherewith
It follows from equation (45) that
Theorem 4. If , , , , , , , and , , where is a positive constant, then decreases as increases.
Proof of Theorem 4. From equation (46) and under the assumptions on parameters, we getwhereBy equation (48), is the larger root of the following quadratic equation:Note thatIt follows from that . Taking partial derivatives with respect to for equation (51) givesMoreover, from the definition of , we haveNote that . Thus, to show , that is to say, , it is sufficient to show for ,We claim that . Sinceit is equivalent to show thatorThis is automatically true since . It follows from thatThus, as required,This completes the proof.
Remark 1. Under the condition of Theorem 4, the following result can be proved similarly; each entry of the matrix decreases in terms of or , and it increases in terms of , , and . Hence, the net reproduction number decreases in terms of or , and it increases in terms of , , and .
4. Persistence and Existence of Positive Equilibria
Under certain conditions, we use the techniques of persistence theory [28, 29] to show the uniform persistence of the predator population and the existence of at least one positive equilibrium (or coexistence equilibrium) when .
Theorem 5. If equation (13) holds and , then system (1) is uniformly persistent, that is, there exists a positive constant such that every solution with satisfiesHence, it has at least one positive equilibrium.
Proof of Theorem 5. LetIt then suffices to prove that (1) is uniformly persistent with respect to .
It is easy to see from Theorem 1 that is positively invariant and system (1) is dissipative. DefineWe claim thatSuppose on the contrary that there exists such that either or , then similar arguments as those in the proof of Theorem 1 gives and for and . This contradicts with the definition of , and hence the claim has been proved.
Next, choose such that . Then, choose such that and . Here, , , and are those defined in the proof of Theorem 2. We claim thatIt is similar to the proof of Theorem 2. By way of contradiction, suppose there exists such that , . Then, there exists such thatThen, satisfies (23) for . As , we have by Lemma 1 and Comparison Theorem. Thus, there exists such that for . It follows that satisfiesfor . Consider the following auxiliary system:Since , we know is an eigenvalue of with a positive eigenvector, say . As , there exists such that . Again, by Comparison Theorem, we have for . It follows that none of and , , is bounded, a contradiction to the boundedness of solutions (see Theorem 1).
Note that is globally asymptotically stable in for equation (17). By the above claim, we know that both and are isolated invariant subsets in , , where is stable manifold of . Clearly, every orbit in converges to either or , and and are acyclic in . By ([30], Theorem 5 and Remark 1), we conclude that system (1) is uniformly persistent with respect to . By [31], system (1) has at least one equilibrium with and . We further claim that . Otherwise, if , by summing up the equilibrium equations for , we get . Thus, , a contradiction. This completes the proof.
5. Simulations
In this section, numerical simulations are used to analyse the effects of the diffusion rates , , and on the net reproduction number in a two-patch environment. Here, for having concrete observation, we let and , .
First, we fix the parameters as follows:
We change the reproduction number of each isolated patch by changing the carrying capacity and the death rate of prey. Table 1 summarizes all the possibilities of the reproduction numbers in both patches.
When , from Figure 2, we find that the net reproduction number increases and then decreases as increases, and the final value of the threshold is greater than unity, but it is not difficult to observe that is almost unchanged if the predator population hardly diffuses (). When the prey population and the predator population simultaneously diffuse, increases and then decreases, and the final value of the threshold is less than unity. In Figure 3, if we reduce the carrying capacity and the death rate of the second isolated patch in Figure 2, that is, decreases from 3 to 2.45 and decreases from 0.23 to 0.15, we draw the conclusion that increases and the final value is greater than unity as increases no matter how changes, and does not change if .
When we increase the production number of the first isolated patch and have , in Figure 4, it is easy to find that the threshold always decreases and is close to unity as increases and . On the contrary, first decreases and then increases and is greater than unity as increases when . However, always decreases and is less than unity as the prey population and the predator population simultaneously diffuse. In Figure 5, we increase the parameters including the carrying capacity and the death rate in the first isolated patch in Figure 4, that is, increases from 2.8 to 3.2 and increases from 0.15 to 0.18. Under these circumstances, we find that the curve of changes a lot. The curve of is waved as increases when and is greater than unity when . always decreases and tends to stabilize as increases when . When the dispersal of the prey and the predator happens simultaneously, the curve of is waved and is finally less than unity.
The situations of Figures 6 and 7 are similar to Figures 4 and 5 owing to the differences of the carrying capacities and the death rates of the preys. Contrary to circumstances of Figures 4 and 5, we know that the final values of in Figures 6 and 7 are greater than unity as .
The situations of Figures 8 and 9 are very interesting; increases and is greater than unity if or , but gradually decrease and is even less than unity when .
6. Discussion
In this paper, we studied a predator-prey model to describe the dynamics of predation with stage structure in the predator and dispersal in an -patch environment. The population dispersal among patches can be interpreted as migration or predation in which the predator has to seek prey by virtue of food reduction. In addition, predation function includes three typical types having biological significance.
In Section 2, we first proposed the predator-prey model with diffusion in an -patch environment. And we proved that the solutions of system (1) are all positive and ultimately bounded with nonnegative initial conditions (Theorem 1). In Section 3, we derived the net reproduction number and obtained the stability of the PFE through the relationship between and 1 (Theorem 2). Moreover, we provided the upper bound and the lower bound of (Theorem 3). For a two-patch model (39), the constant dispersal rate of prey and predator decreases (Theorem 4). If and the dispersal rate matrices of prey and predator are assumed to be irreducible, system (1) persists globally (i.e., in all patches) under some conditions in Section 4 (Theorem 5).
By numerical simulations, we found some interesting phenomena. From Figures 2 and 3, we first know that the net reproduction number may be greater than unity even if , , and it may also be less than unity. This phenomenon is related with the carrying capacities and the death rates of prey . In Figures 4–7, if one of the isolated patch thresholds is greater than unity and the other is less than unity, when the diffusion happens, the curve of changes in a complex way. Here, we still obtain all kinds of figures through changing parameters . It is not hard to think that the curve of becomes complicated when we change the other parameters. In addition, because dispersal leads to extinction of predators, we believe that dispersal may be a bad thing, but sometimes dispersal may be a good thing if we want to eliminate some pests.
There are some interesting questions worthy of further study; for instance, the corresponding conclusions for the above two functional response types still remain open due to the analytical challenge, which remain as future questions.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This study was supported by the NNSF of China (nos. 11871179 and 11771374) and the Fundamental Research Funds of Heilongjiang Provincial Colleges (no. 135309475).