Abstract
In this paper, we consider a predator–prey model with nonuniform predator dispersal, called predation-induced dispersal (PID), which represents predator motility depending on the maximal predation rate and the predator death rate in a spatially heterogeneous region. We study the local stability of the semitrivial steady state when predators are absent for models with PID and linear dispersal. We then investigate the local/global bifurcation from the semitrivial steady state of these models. Finally, we compare the results of the model with PID to the results of the model with linear dispersal. We conclude that the nonuniform dispersal of predators obeying PID increases fitness for predator invasion when rare; thus, predators with PID can invade a region with an increased probability even in cases wherein predators dispersed linearly cannot invade a certain region. Based on the results, we provide an ecological interpretation with the simulations.
1. Introduction
In ecology, species dispersal can occur by various factors such as species density in its habitat, its response to the environment, and interactions with other species causing migration to other habitats. Therefore, a species' response to its habitat environment and interactions with other species are important factors in modelling a more natural dispersal [Citation45,Citation50]. Many authors have studied population models with the evolution of dispersal, depending on other species, represented by reaction–diffusion equations: self and cross-diffusion models [Citation34,Citation39–41] and the prey-taxis model [Citation7,Citation20,Citation25,Citation37,Citation51,Citation53]. Additionally, considerable studies have been performed on models with starvation-driven diffusion that depend on resources [Citation6,Citation8,Citation10,Citation11,Citation30–32]. Migration of species affords such as gaining better-quality resources, fleeing disadvantageous areas, and evading hostile species, which are crucial for species survival. In terms of evolution, the dispersion principle according to environmental conditions suggests how a species evolves to adapt to the environment of the species' habitat and whether this dispersal way provides an evolutionary advantage for the species' survival. Considerable studies have been performed on species dispersal from an evolutionary perspective by establishing discrete and continuous mathematical models [Citation4,Citation12,Citation26,Citation35,Citation36,Citation43–45].
In a heterogeneous habitat, rapid diffusion of species gives a disadvantage to its survival when its dispersal is spatially uniform [Citation17]. In other words, the dispersal of biological organisms in a heterogeneous environment hinders their survival. Thus when establishing a biological model with a spatially heterogeneous habitat, uniform diffusion is inappropriate to represent the biological dispersal of a species. Due to this disadvantageous effect of diffusion to species, it is difficult to construct a proper model that describes the dispersal behaviour of species in habitats with spatial heterogeneity. Particularly in the predator–prey interactions, the effect of dispersal on the environmental conditions in the local habitat has not been well studied. Thus, it is worthwhile to study nonuniform dispersal depending on the location in the habitat to examine which dispersal is evolutionarily advantageous for biological predator–prey species.
Among the main reasons for the dispersal of species is to escape unfavorable regions with food shortages, harsh climate conditions, or pressure from predators for survival. Since species migration is unprofitable to them if the habitat conditions are uniform spatially and temporally, the habitat environment should be considered spatially and temporally heterogeneous. A low dispersal rate benefits species survival when the environment is spatially nonuniform [Citation16,Citation17,Citation24,Citation26,Citation28,Citation42]. On the other hand, a high dispersal rate favors species survival if the environment is changeable over time [Citation16,Citation24,Citation26,Citation42,Citation52]. For a habitat with spatial and temporal heterogeneity, interaction with other species in a region determines the optimal dispersal rate for species survival [Citation16,Citation24]. In previous studies, it has been assumed that species have uniform and random movement, regardless of how the environmental conditions are given. Meanwhile, for the predator–prey model, various types of functional responses to predation have been examined extensively. Many models have a functional response in which ignoring an effect of predator density, i.e. the function that describes the density of prey consumed by its predator depends only on the prey. However, considerable explicit biological and physiological evidence [Citation1,Citation3,Citation13,Citation18] has shown that in various situations, when predators have to search for food in heterogeneous situations, they exhibit a ratio-dependent functional response in which the per capita predator growth rate should be a function of the prey-to-predator abundance ratio [Citation2,Citation4,Citation21–23,Citation27,Citation33,Citation46,Citation48].
In this work, it is assumed that a species tends to stay in a current location when such a location is favorable for survival; thus, the species will make its motility low, as will the dispersal rate. In contrast, if the species is located in an unfavorable region, its motility will be high to escape such a region. In other words, the species will move with rapid dispersal. This movement strategy indicates that even though a species usually does not have prior information on a habitat condition, such motility change generally induces the species to move toward a better environment.
To describe such a phenomenon, we introduce the predator dispersal strategy, predation-induced dispersal (PID), which represents the change in motility of a predator depending on the predation rate and predator death rate in a region. More precisely, if predators are located in a low-predation region wherein prey can easily hide and escape from predators or in a dangerous place where the predator death rate is high, they have to migrate to other regions. In other words, when the predation is low or the predator death rate is high, they disperse faster and more actively and tend to avoid such locations. In contrast, if the predation is high or the predator death rate is low, the predator dispersal is small because the predators are satisfied with their hunting and tend to remain in that region.
In this paper, we suggest a dispersal strategy that cooperates with fitness for predator invasion in a heterogeneous habitat. To achieve our goal, we examine the predator–prey model with ratio-dependent functional responses in which the predator's movement follows the rule of PID to understand how PID affects the fitness for a species invasion. First, we investigate the local stability of the semitrivial steady state when predators are absent for models with PID and linear dispersal. We then study the local/global bifurcation from the semitrivial steady state of these models. Finally, we compare the results of the model with PID to those with linear dispersal. We conclude that the nonuniform dispersal of predators obeying PID increases fitness for predator invasion when rare; thus, predators with PID can invade a region with an increased probability even in cases wherein predators dispersed linearly cannot invade a certain region.
The outline of this article is as follows. In Section 2, we set up the mathematical model in which predators have PID. The stability of the semitrivial steady state of the models with linear dispersal and PID is investigated with ecological interpretations in Section 3. Section 4 is devoted to obtaining the result of the coexistence state of the PID model by using the linearized eigenvalue and local/global bifurcation theory. In particular, the results are compared with those of a model based on linear dispersal. In Section 5, we summarize the obtained results and discuss the biological interpretations with numerical simulations.
2. Mathematical models with PID
In this section, we present a predator–prey model with PID in a spatially heterogeneous environment under no flux boundary conditions. First, we introduce the PID strategy mathematically. Let β and c be the predation rate and predator death rate, respectively; is the maximal predation rate, where m is the half-capturing saturation parameter. Denote a constant i as the measure of satisfaction in the environment defined by It is noteworthy that i increases when the maximal predation is low, and i decreases when the death rate is low. That means that i indicates whether a given environment is good for predator survival. If a predator's habitat is spatially heterogeneous where the predation rate depends on a location x, satisfaction measure i can be extended to function with location variable x as Then, we consider a mobility function , which is an increasing function with respect to i. The increasing property of represents that predators tend to quickly leave the location when they are in an area where is large and stay in the location when they are in an area where is small. At the individual level, the species with the mobility trait decides its moving rate only depending on local habitat conditions, not having a purpose in where to go. Thus, such a decision does not guarantee locating a better place because it does not depend on the arrival point. The dispersal of predator populations with such traits can be represented by . When predators are satisfied with the environment, they reduce their mobility to stay in the region; conversely, when they are in a region where there is not sufficient food to survive, they increase their mobility to avoid such unfavorable locations.
For example, if we consider for some positive constant k, then, which is diffusion with a logarithmic form of advection that represents predators moving toward a high-predation habitat [Citation29].
With the preceding motivations, in this study, we investigate the following predator–prey interaction systems with PID and ratio-dependent functional responses in a spatially heterogeneous environment: (1) (1) where is a bounded domain and the zero-flux condition is given on smooth boundary , where is the outward unit normal vector on the boundary. The functions and represent the prey and predator population densities, respectively, at location x and time t. The predator's reaction term indicates that the predator is fed by only one prey and not by any other resource. Thus, the model describes the interaction by which the predator disperses, , which obeys the PID rule defined below; however, the prey diffuses linearly at a constant rate. Moreover, , , and are positive functions on and represent the carrying capacity, maximal capturing rate by the predator, and maximal predation rate, respectively, in a spatially heterogeneous environment; c,r and m are positive constants that represent the death rate of the predator, the intrinsic growth rate of the prey, and the half-capturing saturation constant, respectively.
In this work, we consider PID under the following assumptions: (G) (G) Assumption (S) will be used for local/global bifurcation of the model. We denote and .
Here is an example of satisfying (G) and (S). If we consider a step function (Figure (a)), as in the simple case of defined by dispersal function can be approximated by a smooth motility function defined by convolution, (2) (2) where is small and is a smooth symmetric mollifier with and . Then, is a smooth function, and converges to as ε approaches 0. Additionally, satisfies (??) and (??) for a sufficiently small .
Now, we introduce a mobility function, , with a shape that satisfies (G) and (S). For the given mobility function , we determine such that: where satisfies (G), (S), and . We call the shape of .
If for a constant ν, predator dispersal is not affected by the surrounding environment. Since for i = 1, the equality implies that the net growth rate is given by: if the prey population is sufficiently large. In this case, the predator need not be concerned about the environment at its location, so it exhibits natural dispersal, ν, independent of the environment at i = 1. Thus, we set to compare the results obtained from the conditions for the PID model to those with a constant dispersal. Therefore, PID is defined as (Figure (b)). Now, we will consider high- and low- predation regions , and , respectively, defined as: The criterion for dividing and is that the maximal predation rate is equal to the death rate, . For expansion, the domain types are defined as follows: Ω is called a high (low) predation domain if Note that if , Ω is an unlivable domain.
The spatial heterogeneity of the environment is represented by a nonconstant function and such spatial heterogeneity is crucial to the stability analysis in this study.
We also consider a predator–prey model with uniform diffusion () in system (Equation1(1) (1) ), where ν is a positive constant: (3) (3) We then analyze the local stability of the semitrivial steady state of (Equation3(3) (3) ) and determine the condition wherein (Equation3(3) (3) ) has a coexistence steady state. The effect of regional heterogeneity and PID on the stability of the semitrivial steady states of model (Equation3(3) (3) ) is investigated. Furthermore, we compare the stability conditions of the model incorporating PID with those of a model with uniform diffusion. In addition, the coexistence steady state of the models is examined for two different dispersal strategies: PID and linear dispersal.
3. Stability of the semitrivial steady state
We consider a predator–prey interaction system (Equation3(3) (3) ) with linear dispersal represented by a constant diffusion and system (Equation1(1) (1) ) with PID. To investigate invasion by predators, we analyze the stability of the semitrivial state and compare the conditions for these two models to examine the effect of PID on fitness for species invasion when rare.
First, we consider the following scalar equation: (4) (4) Let . Then, , and . Since the principal eigenvalue is positive, the single Equation (Equation4(4) (4) ) has a unique positive solution denoted by (see [Citation5]). Therefore, when predators are absent, system (Equation1(1) (1) ) (also model (Equation3(3) (3) )) has a unique semitrivial steady state.
Since initial data and are nonnegative and not identically zero, and for all and all t>0 by the maximum principle [Citation47]. Moreover, are the classical solutions of systems (Equation1(1) (1) ) and (Equation3(3) (3) ) and exist for all times t>0. A positive solution of system (Equation1(1) (1) ) (or (Equation3(3) (3) )), which is , is called the coexistence of system (Equation1(1) (1) ) (or (Equation3(3) (3) )). In addition, we say that a positive steady state of system (Equation1(1) (1) ) (or (Equation3(3) (3) )) is the coexistence state when both components are positive; it is a semitrivial state when one component is positive and the other is zero. Note that in systems (Equation1(1) (1) ) and (Equation3(3) (3) ), there is only one semitrivial steady state .
It is also noteworthy that the prey in system (Equation1(1) (1) ) or (Equation3(3) (3) ) always survives if a certain condition is satisfied.
Theorem 3.1
Let be a solution of (Equation1(1) (1) ) or (Equation3(3) (3) ). If for all , then and .
Proof.
From the first equation of (Equation1(1) (1) ), we have (5) (5) Let be a solution of the equation: (6) (6) From [Citation15], (Equation6(6) (6) ) has a unique positive steady-state solution which is globally asymptotically stable. By the comparison principle and (Equation5(5) (5) ), for . Additionally, we have
3.1. Model with linear dispersal
In this subsection, we present the results of the linear stability for system (Equation3(3) (3) ). Since (Equation3(3) (3) ) is a special case wherein in (Equation1(1) (1) ), we omit the proofs here. We provide proofs for the results of the PID system (Equation1(1) (1) ) in Section 3.2.
In this subsection, we state the results of the stability of the semitrivial steady state. Linearizing system (Equation3(3) (3) ) at , we obtain the following eigenvalue problem: (7) (7) For the linear stability of , we have the following:
Lemma 3.2
If , then is linearly unstable, and if , then is linearly asymptotically stable.
Denote the average of function by : From Lemma 3.2, we have the stability of .
Theorem 3.3
Let be the semitrivial steady-state solution of (Equation3(3) (3) ).
(i) | Suppose that . Then is linearly unstable for any . | ||||
(ii) | Let M be the positive constant satisfying for a given . Then, for a given , there exists depending on , and c such that if , then is linearly asymptotically stable, and if , then is linearly unstable. | ||||
(iii) | For a given ν, there exists depending on and c such that if , then is linearly unstable, and if , then is linearly asymptotically stable. |
According to the above theorem, if either the predator death rate or the half-capturing saturation constant is sufficiently small, predators can invade a region when rare. For a certain range of the half-capturing saturation constant, the threshold value relates to the dispersal rate of the predator and determines the stability of the semitrivial steady state where the predator is absent. In addition, for a given predator dispersal rate, the threshold value relates to the half-capturing saturation constant that determines the stability of the semitrivial steady state where there is no predator. Figure represents such phenomena.
From Theorems 3.1 and 3.3, it is noteworthy that the instability of the semitrivial steady state of (Equation3(3) (3) ) and positivity of imply the coexistence steady states of system (Equation3(3) (3) ). This can be shown using the fixed-point index theory as in [Citation10,Citation29]. Instead, we will provide the coexistence results by using the local/global bifurcations in the next section.
Corollary 3.4
Assume that for .
(i) | If , then (Equation3(3) (3) ) has the coexistence for any . | ||||
(ii) | Let M be a positive constant satisfying for a given . Then, for a given , there exists depending on and c such that if , then (Equation3(3) (3) ) has the coexistence. | ||||
(iii) | For a given ν, there exists such that if , then (Equation3(3) (3) ) has the coexistence. |
3.2. Model with PID
Recall that PID is a dispersal in which a predator's movement is affected by the predation rate in the circumstances of the location in the habitat. More precisely, if a predator favors a certain region since its predation rate is high (this implies that there are good enough prey in the region to satisfy the predator), it tends to stay there. Thus, its motility is low, and it disperses slowly. However, if a predator feels there is a lack of prey, it will avoid that region, and its motility will become high. To represent this phenomenon, PID is defined in Section 1 as follows: (8) (8) where is the shape of mobility function , satisfying (G), (S), and . Here, ν represents the predator dispersal rate independent of the given environmental elements and c. The setting (Equation8(8) (8) ) will be used to compare PID with linear dispersal.
In this subsection, we investigate the stability of the semitrivial steady-state solution of (Equation1(1) (1) ).
Letting , (Equation1(1) (1) ) is equivalent to (9) (9) where .
Denote and By simple calculation, we obtain and the linearized eigenvalue problem at : (10) (10) For the linear stability of , we introduce the lemma, which can be proven similarly to [Citation10,Citation31]:
Lemma 3.5
If , then is linearly unstable, and if , then is linearly asymptotically stable.
Now, we define (11) (11) for a given , c, and m. This ratio R indicates the degree of predation in the predator's living environment by considering the difference between the maximal predation and the predator death rate. For example, if R<1, then the region that represents a strong hunting case is more occupied in the entire domain Ω than a region wherein it is difficult for the predator to hunt prey.
Denote the average of the function by :
Lemma 3.6
Let and be given.
(i) | If , then , | ||||
(ii) | If , there exists such that if , then , and if , then . |
Proof.
(i) follows from the monotonicity of .
(ii) First, we decompose the integral as Since is an increasing function of i, we have Additionally, the continuity of implies that there exists such that Similarly, we can find satisfying Therefore, Hence, by choosing , we get our assertion.
Thus, we obtain the stability of .
Theorem 3.7
Let be the unique semitrivial steady-state solution of (Equation1(1) (1) ), and ρ be the number in Lemma 3.6(ii).
If , then, is linearly unstable for any .
If , then, is linearly unstable for any .
Suppose that , and that M and are given constants in Theorem 3.3(ii). Then, for a given m<M, there exists depending on , and c such that if , then is linearly asymptotically stable, and if , then is linearly unstable.
Let be in Theorem 3.3(iii) for a given ν. Then, for a given p, there exists depending on , and c such that if , then is linearly unstable, and if , then is linearly asymptotically stable.
Proof.
By Lemma 3.5, it suffices to check the sign of . For simplicity, we denote by . By Theorem A.2 and Lemma 3.6, we obtain (i) and (ii).
(iii) Since and have the same sign, the existence of such follows from Theorem A.2. Now, to show that , it suffices to verify that is positive for . Let ϕ be a principal eigenfunction corresponding to with . Then (iv) Let be a principal eigenfunction corresponding to . For given and c, we choose M such that . Then, if m>M, for all . Thus, we have For , is positive. Observing that , we have by Lemma A.1, and thus for . Since is a strictly decreasing function to m, we can find such that for . Now, we only need to show that , where satisfies . Since for , we have follows from the monotonicity of with respect to m.
In Theorem 3.7, as in the model with linear dispersal, if either the predator death rate or the half-capturing saturation rate is low, predators can invade a region. However, for the given half-capturing rate, if the number R is relatively large, then another threshold value exists other than that occurring in the model with linear diffusion (Theorem 3.3) related to predator dispersal such that the stability of the semitrivial steady state changes. Moreover, for the given PID, another threshold value other than that appearing in the model with linear diffusion (Theorem 3.3) relates to the half-capturing rate such that the stability of the semitrivial steady state changes. Figure represents the contents of Theorem 3.7. It shows that if m is less than , then is unstable. However, curve for the linear diffusion case is shifted to the right of curve for PID, which means that the area for the instability of is enlarged in the -plane. Thus there is a range of half-capturing saturation rates m such that the coexistence of system (Equation1(1) (1) ) exists even though (Equation3(3) (3) ) cannot have coexistence. This implies that PID affects the evolution toward fitness for predator invasion in predator–prey interactions. Similarly, we can interpret the results with respect to ν (Theorem 3.7(iii)). The coexistence of system (Equation1(1) (1) ) can be shown from Theorem 3.7 as Corollary 3.4.
Corollary 3.8
Assume that for all and that ρ is the number in Lemma 3.6(ii).
(i) | Suppose that . Then, there is a coexistence of (Equation1(1) (1) ) for any i>0. | ||||
(ii) | Suppose that , and M and are given constants in Theorem 3.3. Then, for given m<M, there exists depending on , and c such that if , then there is a coexistence of (Equation1(1) (1) ). | ||||
(iii) | Let be in Theorem 3.3 for a given ν. Then, for a given p, there exists depending on and c such that if , then there is a coexistence solution of (Equation1(1) (1) ). |
Note that the number is not specified in the preceding results.
3.3. Particular case of
Let us consider the particular form defined in (Equation2(2) (2) ) with where ν is the predator diffusion rate in system (Equation1(1) (1) ). Note that is smooth and satisfies (G) and (S). Thus, we can read Lemma 3.6 for this particular PID as follows:
Lemma 3.9
Let and be given.
(i) | Suppose . Then, ; | ||||
(ii) | Suppose . If , then , and if , then for a sufficiently small , where R is defined in (Equation11(11) (11) ). |
Furthermore, we can rewrite the stability result Theorem 3.7 of system (Equation1(1) (1) ) for PID defined as (Equation2(2) (2) ) (Figure ).
Theorem 3.10
Let be a unique semitrivial steady-state solution of (Equation1(1) (1) ), and ε be small such that Lemma 3.9(ii) holds.
(i) | Suppose that . Then, is linearly unstable for any . | ||||
(ii) | Suppose that , and M and are given constants in Theorem 3.3(ii). Then, for a given m<M, there exists depending on , and c such that if , then is linearly asymptotically stable, and if , then is linearly unstable. | ||||
(iii) | Let be in Theorem 3.3(iii) for a given ν. Then, for a given ℓ and h, there exists depending on , and c such that if , then is linearly unstable, and if , then is linearly asymptotically stable. |
The above theorem indicates that the dispersal sensitivity, , is high, and the instability of the semitrivial state follows; thus, the predators can invade a region when rare, regardless of any PID. However, when the dispersal sensitivity, , for the spatial environment is relatively small, another threshold value other than that occurring in the model with linear diffusion (Theorem 3.3) determines whether the predators can invade a region (see Figure ). In this case, the predators obeying the PID can invade a region when rare, even if they cannot invade when dispersed linearly. Therefore there is a high probability of invading a region compared with the case with linear dispersal. The dispersal sensitivity, , is crucial to determining the invasion of predators in a region. This particular case of interprets the results more evidently than with the general shape of .
4. Coexistence states
4.1. Global/local bifurcation by linear dispersal
We state the instability (coexistence) region for . Denote for a given β, c. From Lemma A.1, we have
Theorem 4.1
For ,
(i) | is decreasing with respect to m and ν. | ||||
(ii) | . | ||||
(iii) | . Furthermore, . |
Note that combining the above theorems, we obtain a continuous curve (Figure ) : Now, we obtain the global/local bifurcation and coexistence state of (Equation3(3) (3) ): (12) (12) First, we find the a priori bound of (Equation12(12) (12) ).
Lemma 4.2
Assume that is a positive solution of (Equation12(12) (12) ). Then,
Assume that , , , and c are given. For all values of m, we have the branch of trivial solutions and of semitrivial steady states to elliptic system (Equation12(12) (12) ). By applying the local bifurcation theorem of Crandall and Rabinowitz [Citation14], we obtain a branch of positive solutions that bifurcates from . For p>n, we define Banach spaces X and Y as where . Then, by the Sobolev embedding theorem, , where is a set of continuously differentiable functions with compact support in Ω.
From Theorems A.2 and 3.3, there exists such that . Thus, there is the positive function which is a solution of the equation: where .
Choosing as a bifurcation parameter, we obtain the local bifurcation property as follows.
Lemma 4.3
Assume that , and c are given. Then, a branch of positive solutions to elliptic Equation (Equation12(12) (12) ) bifurcates from the semitrivial steady state solution curve if and only if . In other words, all positive solutions of the elliptic equation near can be represented by where . Here, is a smooth function with respect to s and satisfies and .
Before we investigate the global nature of the solution curve in Lemma 4.3, we give the bound of the values m for the existence of positive solutions to (Equation12(12) (12) ).
Theorem 4.4
If is a nontrivial, nonnegative solution of (Equation12(12) (12) ), then .
The global bifurcation result for parameter m is obtained as follows:
Theorem 4.5
Assume that , and c are given. If , then (Equation12(12) (12) ) has at least one positive solution. The set of positive solutions of (Equation12(12) (12) ) with respect to bifurcation parameter m forms a continuum satisfying the following properties:
(i) | bifurcates from at , | ||||
(ii) | . |
4.2. Global/local bifurcation by PID
We investigate the global/local bifurcation and coexistence of the time-independent system of (Equation1(1) (1) ): (13) (13) where and . We regard as a bifurcation parameter. First, we obtain the a priori bound of (Equation13(13) (13) ).
Lemma 4.6
Assume that is a positive solution of (Equation13(13) (13) ). Then
Proof.
Let be points satisfying and . By the maximum principle, and . From this, we have Thus, . Similarly, Therefore, .
We assume that , , , , and c are given. For all values of m, we have the branch of trivial solutions and of semitrivial steady states of elliptic system (Equation13(13) (13) ). Using a similar argument as in Section 4.1, we can obtain the branch of positive solutions that bifurcates from .
From the proof of Theorem 3.7, for , and there is a positive function satisfying where .
We have the following local bifurcation property:
Lemma 4.7
Assume that , and c are given. Then, the branch of positive solutions of elliptic Equation (Equation13(13) (13) ) bifurcates from the semitrivial steady state solution curve if and only if . In other words, all positive solutions of the elliptic equation near can be expressed by where , is a smooth function with respect to s and satisfies and .
Proof.
Let . Define the operator by Since is the semitrivial steady state of elliptic Equation (Equation13(13) (13) ), we have . By simple calculations, we obtain Then, , where in Ω. If , then there exists such that (14) (14) By the Fredholm alternative theorem, the second equation of (Equation14(14) (14) ) is solvable if and only if . For such a solution ψ, the first equation has a unique solution ϕ since is invertible. Thus, by the properties of the compact operator, Furthermore, we have Since by (S), we have Consequently, we can apply the local bifurcation theorem in [Citation14] to F at . Note that there are no other bifurcation points except by virtue of the Krein–Rutman theorem.
Next, we give the bound of values, m, for the existence of positive solutions of (Equation13(13) (13) ).
Theorem 4.8
If is a nontrivial, nonnegative solution of (Equation13(13) (13) ), then .
Proof.
Suppose that . From (Equation13(13) (13) ) of the V-equation with , Thus, we obtain Multiplying V by both sides and integrating on Ω, we have On the other hand, from the result in Theorem 3.7, when . Therefore, which is a contradiction. Hence, .
The global bifurcation structure is obtained as follows:
Theorem 4.9
Let , and c be given. Assume that for all . Then, (Equation13(13) (13) ) has at least one positive solution if and only if . The set of positive solutions of (Equation13(13) (13) ) for bifurcation parameter m forms continuum satisfying the following properties:
(i) | bifurcates from at , | ||||
(ii) | . |
Proof.
For the local bifurcation branch from Lemma 4.7, we denote the maximum connected set of . First, we introduce the positive cone: We claim that . Suppose that satisfies . Then, it follows that for , and so By the strong maximum principle, or .
We suppose that . Then, lies on the trivial branch of solutions . Let be the sequence of positive solutions of (Equation13(13) (13) ) with in . By Harnack's inequality, there exists such that . By dividing the first equation of (Equation13(13) (13) ) by and integrating over Ω, we have (15) (15) (16) (16) where . Note that for sufficiently large n. Therefore, for large n, which the second integration (Equation15(15) (15) ) is positive, and so it is a contradiction. Thus, there is no sequence converging to . Therefore, either or should be nonzero. Suppose that is zero. Then, . However, since this equation cannot have a nonzero solution, is not identically 0. Suppose that is zero. Since is the only bifurcation point, , which is impossible. Therefore, .
Next, by global bifurcation theory [Citation38,Citation49], one of the following nonexcluding properties holds:
is unbounded in ;
meets the semitrivial steady state set at another bifurcation point, except ;
connects to another component on the boundary of .
From the a priori bound of solutions of (Equation13(13) (13) ) (Lemmas 4.6) and 4.8, (i) cannot hold. Additionally, (ii) cannot occur since the positive solutions bifurcate from the semitrivial steady state solution curve only at by Lemma 4.7. Hence, alternative (iii) must hold. Then, the positive solution of (Equation13(13) (13) ) on the boundary of has the form , and . Since is a unique bifurcation point and is unstable for any m under the assumption for all , is the only possible form of the positive solution. Therefore, .
Figure is the bifurcation diagram representing the curve of the density of predators from the semitrivial state for a system with random dispersal and PID of predators, as half-capturing saturation constant m is a bifurcation parameter. The bifurcation curves begin at threshold values and with , which occur in Theorems 3.3(iii) and 3.7(iv), respectively. Thus, there is a range of m such that , which indicates that if the predators have motility obeying PID, they can coexist with the prey even though there is no coexistence when the predators diffuse linearly.
Furthermore, we can take ν as a bifurcation parameter for systems (Equation1(1) (1) ) and (Equation3(3) (3) ).
The bifurcation diagram in Figure is provided for a bifurcation parameter ν. The curves start at two distinct threshold values, and , from Theorems 3.3(ii) and 3.7(iii), respectively. Thus, as in the case where m serves as a bifurcation parameter, there are values ν such that . This result again shows that when the predators move following PID, the coexistence state exists even though the predators cannot survive with the prey when the former diffuses linearly. Since the results are similar to the results obtained when we take m as a parameter, we present them in Appendix 2.
5. Conclusion
In ecology, the motility of species in a certain region depends on the circumstances of the local habitat. In this research, we studied the evolution of dispersal toward fitness for the invasion of predators, called PID as a dispersal strategy by examining the effect of PID on the stability of semitrivial steady states. First, we considered a predator–prey model with linear dispersal. If the predator death rate or the half-capturing saturation rate is low for any dispersal rate of the predator, the predator can invade a region when rare. Moreover, when the half-capturing saturation constant is in a particular range, the threshold value relates to the predator dispersal rate, which determines the stability of the prey where the predator is absent. Furthermore, for a given predator dispersal rate, the threshold value is related to the half-capturing saturation constant that changes the prey's stability where the predator is absent. In addition, the instability of the semitrivial steady state implies the coexistence of the time-dependent system from the positive density of the prey because there is only one semitrivial steady state, (Figure ). Whereas, when the predator follows PID, if the predator death rate or the predation degree is sufficiently low for any PID of the predator, it can invade a region when rare. In addition, given the half-capturing rate, when the predation degree is relatively high, another threshold value other than that occurring in the model with linear dispersal relates to the dispersal rate of the predator such that the stability of the semitrivial steady state changes. Furthermore, for a given PID, another threshold value other than that occurring in the model with linear dispersal relates to the half-capturing rate such that the stability of the semitrivial steady state is changeable (Figure ). Thus, the predator has a higher probability of invading a region than in the case with linear dispersal (Figure ).
In particular, when PID is defined as the shape of a step function with bounds h and ℓ, we obtain more applicable conditions for stability. More precisely, when the dispersal sensitivity is high, the predator can invade a region, regardless of any PID (Figure ). That is, the high sensitivity of predator dispersal always makes predators survive even in a low predation domain (Figure ). In contrast, if the dispersal sensitivity is relatively low, another threshold value other than that occurring in the model with linear dispersal determines the invasion of the predator in a region (Figure ). In this case, the predator with PID can invade when rare in a region if it cannot invade when dispersed linearly (Figures and ). Thus, dispersal sensitivity is crucial in determining the invasion of predators in a region for the PID of step-function PID. We can conclude that the evolution of dispersal toward fitness for predator invasion to a given region for predators progresses effectively when their movement obeys PID rather than linear dispersal.
Finally, we obtained the coexistence results of the time-independent system obtained using a bifurcation branch, starting a unique semitrivial steady state of models with a linear diffusion and PID, also supporting that the PID of predators can increase fitness for invasion of the predators when rare.
In this paper, we adopted the ratio-dependent functional response to describe the predator–prey interaction. If the prey-dependent functional response, such as Holling type II, is considered, we can see a similar effect of PID on predator survival, but it would be more complicated, and some different results from those of this paper, depending on the prey's conditions, will also be obtained. Finally, we note that the uniqueness and stability of the coexistence state is an important issue in the mathematical ecology model, especially for predator–prey models in a spatially heterogeneous environment. We leave this challenging issue open and hope to be solved in future works.
Acknowledgments
The authors greatly appreciate the valuable comments and suggestions of anonymous reviewers of this paper and the editor, which significantly improved the original version.
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References
- R. Arditi, L.R. Ginzburg, and H.P. Akcakaya, Variation in plankton densities among lakes: A case for ratio-dependent predation models, Am. Nat. 138 (1991), pp. 1287–1296.
- R. Arditi and L.R. Ginzburg, Coupling in predator–prey dynamics: Ratio-dependence, J. Theor. Biol.139 (1989), pp. 311–326.
- R. Arditi and H. Saiah, Empirical evidence of the role of heterogeneity in ratio-dependent consumption, Ecology 73 (1992), pp. 1544–1551.
- R.S. Cantrell and C. Cosner, On the dynamics of predator–prey models with the Beddington–DeAngelis functional response, J. Math. Anal. Appl. 257 (2001), pp. 206–222.
- R.S. Cantrell and C. Cosner, Spatial Ecology via Reaction–Diffusion Equations, John Wiley & Sons, 2004.
- E. Cho and Y.J. Kim, Starvation driven diffusion as a survival strategy of biological organisms, Bull. Math. Biol. 75 (2013), pp. 845–870.
- W. Choi and I. Ahn, Effect of prey-taxis on predator's invasion in a spatially heterogeneous environment, Appl. Math. Lett. 98 (2019), pp. 256–262.
- W. Choi and I. Ahn, Strong competition model with non-uniform dispersal in a heterogeneous environment, Appl. Math. Lett. 88 (2019), pp. 96–102.
- W. Choi and I. Ahn, Predator invasion in predator–prey model with prey-taxis in spatially heterogeneous environment, Nonlinear Anal. Real World Appl. 65 (2020), pp. 103495.
- W. Choi and I. Ahn, Predator–prey interaction systems with non-uniform dispersal in a spatially heterogeneous environment, J. Math. Anal. Appl. 485 (2020), pp. 123860.
- W. Choi, S. Baek, and I. Ahn, Intraguild predation with evolutionary dispersal in a spatially heterogeneous environment, J. Math. Biol. 78 (2019), pp. 2141–2169.
- D. Cohen and S.A. Levin, Dispersal in patchy environments: The effects of temporal and spatial structure, Theor. Popul. Biol. 39 (1991), pp. 63–99.
- C. Cosner, D.L. DeAngelis, J.S. Ault, and D.B. Olson, Effects of spatial grouping on the functional response of predators, Theor. Popul. Biol. 56 (1999), pp. 65–75.
- M.G. Crandall and P.H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), pp. 321–340.
- D.L. DeAngelis, W.M. Ni, and B. Zhang, Dispersal and spatial heterogeneity: Single species, J. Math. Biol. 72 (2016), pp. 239–254.
- U. Dieckman, B. OaHara, and W. Weisser, The evolutionary ecology of dispersal, Trends Ecol. Evol.14 (1999), pp. 88–90.
- J. Dockery, V. Hutson, K. Mischaikow, and M. Pernarowski, The evolution of slow dispersal rates: A reaction diffusion model, J. Math. Biol. 37 (1998), pp. 61–83.
- A.P. Gutierrez, Physiological basis of ratio-dependent predator–prey theory: The metabolic pool model as a paradigm, Ecology 73 (1992), pp. 1552–1563.
- X. He and W.M. Ni, Global dynamics of the Lotka–Volterra competition–diffusion system: Diffusion and spatial heterogeneity I, Commun. Pure Appl. Anal. 69 (2016), pp. 981–1014.
- X. He and S. Zheng, Global boundedness of solutions in a reaction–diffusion system of predator–prey model with prey-taxis, Appl. Math. Lett. 49 (2015), pp. 73–77.
- S.B. Hsu, T.W. Hwang, and Y. Kuang, Global analysis of the Michaelis–Menten-type ratio-dependent predator–prey system, J. Math. Biol. 42 (2001), pp. 489–506.
- S.B. Hsu, T.W. Hwang, and Y. Kuang, Rich dynamics of a ratio-dependent one-prey two-predators model, J. Math. Biol. 43 (2001), pp. 377–396.
- S.B. Hsu, T.W. Hwang, and Y. Kuang, A ratio-dependent food chain model and its applications to biological control, Math. Biosci. 181 (2003), pp. 55–83.
- V. Hutson, K. Mischaikow, and P. Polácik, The evolution of dispersal rates in a heterogeneous time-periodic environment, J. Math. Biol. 43 (2001), pp. 501–533.
- H.Y. Jin and Z.A. Wang, Global stability of prey-taxis systems, J. Differ. Equ. 262 (2017), pp. 1257–1290.
- M. Johnson and M. Gaines, Evolution of dispersal: Theoretical models and empirical tests using birds and mammels, Annu. Rev. Ecol. Syst. 21 (1990), pp. 449–480.
- C. Jost, O. Arino, and R. Arditi, About deterministic extinction in ratio-dependent predator–prey models, Bull. Math. Biol. 61 (1999), pp. 19–32.
- M Keeling, Spatial models of interacting populations, in Advanced Ecological Theory: Principles and Applications, J. McGlade, ed., Blackwell Science, Oxford, 1999
- K. Kim and W. Choi, Local dynamics and coexistence of predator–prey model with directional dispersal of predator, Math. Biosci. Eng. 17(6) (2020), pp. 6737–6755.
- Y.J. Kim and O. Kwon, Evolution of dispersal with starvation measure and coexistence, Bull. Math. Biol. 78 (2016), pp. 254–279.
- Y.J. Kim, O. Kwon, and F. Li, Evolution of dispersal toward fitness, Bull. Math. Biol. 75 (2013), pp. 2474–2498.
- Y.J. Kim, O. Kwon, and F. Li, Global asymptotic stability and the ideal free distribution in a starvation driven diffusion, J. Math. Biol. 68 (2014), pp. 1341–1370.
- Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator–prey system, J. Math. Biol. 36 (1998), pp. 389–406.
- K. Kuto and Y. Yamada, On limit systems for some population models with cross-diffusion, Discrete Contin. Dyn. B 17 (2012), pp. 2745–2769.
- K.Y. Lam and Y. Lou, Evolutionary stable and convergent stable strategies in reaction–diffusion models for conditional dispersal, Bull. Math. Biol. 76 (2014), pp. 261–291.
- K.Y. Lam and W.M. Ni, Limiting profiles of semilinear elliptic equations with large advection in population dynamics, Discrete Contin. Dyn. Syst. 28 (2010), pp. 1051–1067.
- C. Li, X. Wang, and Y. Shao, Steady states of a predator–prey model with prey-taxis, Nonlinear Anal.97 (2014), pp. 155–168.
- J. Lopez-Gomez, Global bifurcation for Fredholm operators, Rend. Istit. Mat. Univ. Trieste. 48 (2016), pp. 539–564.
- Y. Lou and W.M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differ. Equ. 131 (1996), pp. 79–131.
- Y. Lou, W.M. Ni, and S. Yotsutani, On a limiting system in the Lotka–Volterra competition with cross-diffusion, Discrete Contin. Dyn. Syst. 10 (2004), pp. 435–458.
- Y. Lou, Y. Tao, and M. Winkler, Nonexistence of nonconstant steady-state solutions in a triangular cross-diffusion model, J. Differ. Equ. 262 (2017), pp. 5160–5178.
- M. McPeek and R. Holt, The evolution of dispersal in spatially and temporally varying environments, Am. Nat. 140 (1992), pp. 1010–1027.
- T. Nagylaki, Introduction to Theoretical Population Genetics, Biomathematics Vol. 21, Springer, Berlin, 1992.
- W.M. Ni, The Mathematics of Diffusion, Vol. 82, SIAM, 2011.
- A. Okubo and S.A. Levin, Diffusion and Ecological Problems: Modern Perspectives, Vol. 14, Springer Science & Business Media, 2013.
- P.Y. Pang and M. Wang, Qualitative analysis of a ratio-dependent predator–prey system with diffusion, P. R. Soc. Edinb. A 133 (2003), pp. 919–942.
- M.H. Protter and H.F. Weinberger, Maximum Principles in Differential Equations, Springer Science & Business Media, 2012.
- K. Ryu and I. Ahn, Positive solutions for ratio-dependent predator–prey interaction systems, J. Differ. Equ. 218 (2005), pp. 117–135.
- J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differ. Equ. 246 (2009), pp. 2788–2812.
- J.G. Skellam, Some philosophical aspects of mathematical modeling in empirical science with special reference to ecology, Math. Models Ecol. 13 (1972), pp. 13–28.
- Y. Tao, Global existence of classical solutions to a predator–prey model with nonlinear prey-taxis, Nonlinear Anal. 11 (2010), pp. 2056–2064.
- J.M.J. Travis and C. Dytham, Habitat persistence, habitat availability and the evolution of dispersa, Proc. R. Soc. Lond. B 266 (1999), pp. 723–728.
- S. Wu, J. Shi, and B. Wu, Global existence of solutions and uniform persistence of a diffusive predator–prey model with prey-taxis, J. Differ. Equ. 260 (2016), pp. 5847–5874.
Appendices
Appendix 1.
Eigenvalue problem
We give some properties of the following eigenvalue problem that will be mainly used to obtain the stability results: (A1) (A1) We call the largest eigenvalue of (EquationA1(A1) (A1) ) the principal eigenvalue, denoted by , and defined as Then, there exists the positive principal eigenfunction, , corresponding to .
We introduce the following lemmas about the well-known property of the principal eigenvalue and some conditions for determining the sign of the principal eigenvalue . We omit the proofs here; see [Citation9,Citation19].
Lemma A.1
(i) | is monotonically decreasing with respect to ν. | ||||
(ii) | for . | ||||
(iii) | . | ||||
(iv) | . |
Theorem A.2
(i) | If , then . | ||||
(ii) | If , then . | ||||
(iii) | If and is positive somewhere in Ω, then there exists such that if and only if . The equality holds when . |
Appendix 2.
Global bifurcation by parameter ν
A.1. Model with linear diffusion
We choose as a bifurcation parameter. Then, we obtain a bifurcation curve from the semitrivial steady state of elliptic Equation (Equation12(12) (12) ) by considering ν as a bifurcation parameter. From Theorem 3.3, there are no bifurcation phenomena for and . Thus, we assume that . Then, we have the corresponding results to the above case in which m acts as a bifurcation parameter when m is replaced by ν.
From Theorems A.2 and 3.3, for : is the solution of the equation where .
The local bifurcation property holds as follows.
Lemma A.3
Assume that , and c are given, and . Then, a positive solution of elliptic Equation (Equation12(12) (12) ) bifurcates from the semitrivial steady state solution curve if and only if . In other words, all positive solutions of (Equation12(12) (12) ) near can be expressed by where . Here, is a smooth function with respect to s and satisfies and .
We state the global bifurcation result for system (Equation12(12) (12) ) with the bound of such that (Equation12(12) (12) ) has a coexistence steady state.
Theorem A.4
Let , and be given. Assume that for all . Then (Equation12(12) (12) ) has at least one positive solution if and only if . The set of positive solutions of (Equation12(12) (12) ) with bifurcation parameter ν forms a continuum satisfying the following:
(i) | bifurcates from at , | ||||
(ii) | . |
A.2. Model with PID
Next, we investigate the bifurcation curve on the semitrivial steady state set of elliptic Equation (Equation13(13) (13) ) by considering ν as a parameter, where δ is a shape of p such that . From Theorem 3.7(iv), there are no bifurcation phenomena for or . Thus, we assume that . Consider the equivalent form of (Equation13(13) (13) ) (A2) (A2) Regarding ν as a bifurcation parameter, we have the branch of trivial solutions, and the branch of semitrivial steady states, , of elliptic system (EquationA2(A2) (A2) ). Then, as before, we obtain a branch of positive solutions that bifurcates from . For p>n, we define Banach spaces X and Y as where . By the Sobolev embedding theorem, .
From the proof of Theorem 3.7, . There is a positive function is a solution of the equation where .
The following local bifurcation property holds.
Lemma A.5
Assume that μ, K, α, β, and c are given, and . The branch of positive solutions of elliptic Equation (EquationA2(A2) (A2) ) bifurcates from the semitrivial steady state solution curve if and only if . In other words, all positive solutions of (EquationA2(A2) (A2) ) near can be represented by where . Here, is a smooth function with respect to s and satisfies and .
There is a bound of values ν for the existence of positive solutions of (EquationA2(A2) (A2) ).
Theorem A.6
If is a nontrivial, nonnegative solution of (EquationA2(A2) (A2) ), then .
Proof.
Suppose that . Then, from the second Equation (EquationA2(A2) (A2) ), we have Multiply v by both sides and integration, then From the result in Theorem 3.7, when . Therefore, and so we have a contradiction. Hence, .
Finally, we obtain the following global bifurcation result of (Equation13(13) (13) ) for parameter ν as a similar argument of Theorem 4.9.
Theorem A.7
Let , and be given. Assume that for all . Then (Equation13(13) (13) ) has at least one positive solution if and only if . The set of positive solutions of (Equation13(13) (13) ) with bifurcation parameter ν forms continuum satisfying the following properties:
(i) | bifurcates from at , | ||||
(ii) | . |