Complexity
Volume 2021 (2021), Article ID 9970255, 10 pages
https://doi.org/10.1155/2021/9970255

Research Article

Global Bifurcation Structure of a Predator-Prey System with a Spatial Degeneracy and B-D Functional Response

Xiaozhou Feng,¹ Changtong Li,¹ Hao Sun,1,2 and Yuzhen Wang¹

¹School of Science, Xi’an Technological University, Xi’an 710032, China
²Test Measuring Academy of Norinco, Huayin 714200, China

Correspondence should be addressed to Xiaozhou Feng; flxzfxz8@163.com

Received 26 March 2021; Accepted 2 June 2021; Published 21 June 2021

Academic Editor: Abdel-Haleem Abdel-Aty

Copyright © 2021 Xiaozhou Feng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, we investigate a predator-prey system with Beddington–DeAngelis (B-D) functional response in a spatially degenerate heterogeneous environment. First, for the case of the weak growth rate on the prey (), a priori estimates on any positive steady-state solutions are established by the comparison principle; two local bifurcation solution branches depending on the bifurcation parameter are obtained by local bifurcation theory. Moreover, the demonstrated two local bifurcation solution branches can be extended to a bounded global bifurcation curve by the global bifurcation theory. Second, for the case of the strong growth rate on the prey (), a priori estimates on any positive steady-state solutions are obtained by applying reduction to absurdity and the set of positive steady-state solutions forms an unbounded global bifurcation curve by the global bifurcation theory. In the end, discussions on the difference of the solution properties between the traditional predator-prey system and the predator-prey system with a spatial degeneracy and B-D functional response are addressed.

1. Introduction

The population dynamics system is the basic model to study the spatial and the temporal structures of the biological population. It is used to describe the dynamic distribution of population density produced by the interaction of the species in the ecosystem or the surrounding environment. In particular, the interaction of the predator-prey system is one of the basic structures in complex ecosystems, and such models have been widely studied (see [1–17]) under the uniform condition of the space, in other words, all coefficients of the model are positive. Because the natural environment of most species is spatially heterogeneous, biologists and mathematicians believe that the inhomogeneity of the spatial environment has a significant impact on the dynamic behavior of the biological population system, which is confirmed by the biologist C. B. Huffaker’s biological experiment. To investigate the effect of spatial heterogeneity on the dynamic behavior of the biological population system, a natural way is to replace the constant in the model with a function containing spatial variables. Therefore, we will study a predator-prey system with spatial degeneracy and B-D-type response function [1, 2].

where is the outward unit normal vector of the boundary in . and are the densities of prey and predator, respectively. and are the intrinsic growth rate of prey and predator , respectively. and are capturing rate to predator and conversion rate of prey captured by a predator, respectively. denotes the saturation coefficient, and is the inhibition of functional response function on behalf of predators. The parameters are assumed to be only positive constants, while can be negative. Let be a subset of . Assume that in , and in , where is a function dependent on space variables , which means that two species live in a heterogeneous environment.

When the space environment is homogeneous, that is, is a positive number in , system (1) is reduced to the constant-coefficient system, and the predator-prey system with Beddington–DeAngelis-type functional response has been studied by many scholars from different perspectives [3–5].

In this paper, the different behaviors of system (1) will be mainly discussed under the condition of spatial heterogeneity. If is a positive function in , then there is no essential difference between the model and previous research. If degenerates partially into zero within and , the research results of system (1) show that the behavior of solutions has changed substantially. More concretely, when , system (1) exhibits some fixed constant such that ; then the behavior of the system (1) is similar to the case of as the normal number. For , the behavior of system (1) has undergone an essential change, which means that even if the mortality of predators is very large, the system (1) still has a stable positive solution. Moreover, the asymptotic behavior of the coexistence solution of the equilibrium state on the system (1) was discussed in detail in references [10, 18] as .

For the case where , there is no relevant research result. Therefore, the main purpose of this paper is to investigate the change rule of solutions if partially degenerates to zero in and . Is there a fundamental change in the behavior of system (1)? Are these changes the same as the case of ? Answers to these questions are detailed in the “Conclusion” section.

This paper is organized as follows. In Section 2, some notations and important lemmas used in previous papers is given, including the generalized comparison principle and the convergence property of the semitrivial solution . In Section 3.1, the system (1) with the weak growth rate on the prey () is studied; a priori estimates on any positive steady-state solutions are established by the comparison principle; two local bifurcation solution branches depending on the bifurcation parameter are obtained by local bifurcation theory, and the demonstrated two local bifurcation solution branches can be extended to a bounded global bifurcation curve by the global bifurcation theory. In Section 3.2, the system (1) with the strong growth rate on the prey () is investigated; a priori estimates on any positive steady-state solutions are obtained by applying reduction to absurdity; and then we prove that the set of positive steady-state solutions forms an unbounded global bifurcation curve by the global bifurcation theory. In Section 4, some discussions on the difference of the solution properties between the traditional predator-prey system and the predator-prey system with a spatial degeneracy and B-D functional response are listed.

2. Preliminaries

The main purpose of this section is to give some important lemmas and notations used in later papers.

In this paper, we always suppose that is a nonnegative function in ; moreover, there exists a subset such that and . The research of this paper is dependent on the above assumption and is not suitable for the case .

Obviously, satisfies the second equation of system (1). For this case, satisfies the following logistic equation:

According to the result of [19], we know that Equation (2) has a unique zero solution if ; Equation (2) exhibits the unique zero solution denoted by if . Moreover, the mapping is strictly monotone increasing with .

Next, let be the principal eigenvalues of the operator under the Dirichlet boundary condition on region . For convenience, we introduce the notation , which stands for the principal eigenvalue of the following eigenvalue problem:

Under the sense of these signs, , By the conclusion of [19], in the sense of the norm , we obtain that if , then . If ,

where represents the minimal positive solution of the following boundary value blow-up problem:

where implies .

In a word, if , then system (1) exhibits a unique semitrivial solution . For , system (1) does not exhibit such a semitrivial solution.

If , then satisfies the following logistic equation:

By [20], if , this equation does not have a positive solution. If , this equation has a unique positive solution denoted by . Hence, if , then system (1) has a unique semitrivial solution .

To end this paper, we need to introduce some results. First, we introduce the following generalized comparison principle.

Lemma 1. (see [19]). If are two order continuous derivable positive functions in , and satisfy

then satisfy in . Next, let’s introduce a very useful lemma, which plays a key role in proving a priori estimates of positive solutions.

Lemma 2. (see [10]). Suppose satisfies

where is a positive constant. Then there exists a function such that weakly converges to with the sense of norm; strongly converges to with the sense of norm .

In a paper [21], López-Gómez and Sabina de Lis have proved that converges uniformly to with on any compact subset of . Next, we demonstrate the following result for the case of .

Lemma 3. Let be the corresponding eigenfunction to and . If in , then

Proof.  

Let , then and satisfies

By multiplying two sides of Equation (10) by and integrating over by parts, we get

Hence, as , is uniformly bounded in ; it follows that weakly converges to depending on the norm of , and strongly converges to depending on the norm of . Since , we know that , then . We chose any function , and by multiplying two sides of Equation (10) by and integrating over by parts, we obtain

Let , then , which implies that is a weak solution of the following equation:

According to Harnack’s inequality, , then . Thanks to the regularization theory, with under the norm of . Equation (10) can be rewritten as follows:

By multiplying two sides of the above equation by and integrating over by Green’s formula, we obtain

Let , then . Thus, as , we have

This completes the proof.

3. Global Bifurcation Structure

In this section, we will discuss the global bifurcation structure of system (1) and give sufficient conditions for the coexistence of the two species. Let be fixed, we will investigate the global bifurcation of system (1) by dividing two different cases as follows:

At the same time, unless specifically explained, parameters are fixed. The parameter will be considered as the bifurcation parameter. Next, we will discuss two cases depending on the value range of by two subsections.

3.1. Weak Growth Rate on the Prey ()

For any , system (1) has two semitrivial solutions and . Hence, system (1) has two semitrivial positive curves as follows:

To get our conclusion, we first have to give a priori estimates on any positive steady-state solutions by the comparison principle as follows.

Lemma 4. Suppose that . Any positive solution of system (1) satisfies

where is the unique solution of Equation (2). Moreover, if system (1) has a positive solution, then the parameter satisfies

Proof. Notice that the first equation of system (1), it is easy to get

It follows from the comparison principle that , where is the unique solution of (2). Similarly, according to the second equation of (1), we directly get

Combining with the comparison principle, we obtain

If system (1) has a positive solution, we easily get

Next, two local bifurcation solution branches depending on the bifurcation parameter are obtained by local bifurcation theory as follows. Let . The operator can be defined as follows:

Next, we prove that is a local bifurcation point of system (1), where . For simplicity, let . By direct calculation, we obtain

For , it is easy to get , where satisfies

Since , we can choose . On the other hand, is a positive solution of Equation (2), which demonstrates that is a positive operator, and . The range can be represented as , since , we have . Hence, by the local bifurcation theory [20], we obtain the following result on the local bifurcation solution.

Theorem 1. Suppose that , the local positive bifurcation solution set that bifurcates from the point of system (1) forms a smooth curve

where . By calculating, we get

where is a linear functional in space , defined as follows.

Similarly, we can prove that

is also a local bifurcation point of system (1), where is determined uniquely by .

Set , define the operator by

Let . By directly calculating the Fréchet derivative of the operator, we get

It follows that

For , it is easy to get , where satisfies

Since , we can choose . On the other hand, is a positive operator, and . The range can be represented as , since , it follows that

Thus, according to the local bifurcation theory [22], we obtain similar result as follows.

Theorem 2. Suppose that , the local bifurcation branch near of system (1) forms a smooth curve

where . By calculating, we get

where is a linear functional in space , defined as follows:

Next, we use the modified global bifurcation theorem [23] to prove the global bifurcation structure of the system (1) under the condition of the weak growth rate on the prey.

Theorem 3. Suppose that , then the positive solution set of system (1) forms a bounded smooth curve that connects and and satisfies

where . Moreover, the bifurcation direction of at point is supercritical (). If , then the bifurcation direction of at point is supercritical (); if , then the bifurcation direction of at point is subcritical (), where is determined by

Proof. The proof of this theorem is similar to Theorem 4 below; the detailed proofs are omitted here.

3.2. Strong Growth Rate on the Prey ()

Comparing with the weak growth rate on the prey, there exists only one semitrivial solution curve for the system (1) under the strong growth rate on the prey. In this case, is still a local bifurcation point. The positive solution near of system (1) forms a smooth curve . By the method of Lemma 4, we can prove that if system (1) has positive solutions. Next, we establish a boundary result of any positive solutions of system (1) when is bounded.

Lemma 5. If , then there exists a positive constant that does not depend on such that any positive solution of system (1) satisfies

Proof. Since , it follows that

Hence, . Suppose the conclusion of the lemma does not hold. Then for , there exists some positive solution sequence of system (1) such that with .
Set . The first equation of system (1) implies . Following Lemma 2, weakly converges to in and strongly converges to in , and . Similarly, let . The first equation of system (1) implies

Hence, weakly converges to in and strongly converges to in , and .
Next, it turns out that is almost zero in . According to Lemma 1, it is easy to see that in . Hence, is uniformly bounded on an arbitrary subset of . Thanks to , so is uniformly convergent to 0 on any subset of . Thus, is almost zero in the . And because is smooth enough, then .
According to the first equation of system (1), we have

By multiplying two sides of the above equation by and integrating over , where the support set of the function is , we obtain

Letting , we get

But as ,

Hence, is a weak solution to the following problem:

Thus, by using Harnack’s inequality, we know that or in . If , then ; this is in contradiction with . If , then in ; this is in contradiction with . Therefore, it follows that is uniformly bounded by the antievidence method.

Theorem 4. Suppose that , then the positive solution set of system (1) forms an unbounded smooth curve that extends to by . If , then the bifurcation direction of at point is supercritical (); if , then the bifurcation direction of at point is subcritical (), where is defined in Theorem 3.

Proof. In order to apply the global bifurcation theorem of [23] better, we define the following mapping by

According to the regularization theory of elliptic equations and the Sobolev embedding theorem, the second term of the mapping is a compact mapping. Moreover, the nonnegative solution of the system (1) is equivalent to the zero point of the mapping .
The Fréchet derivative of the mapping at is given as follows:

The corresponding adjoint operator is recorded as , that is,

By standard calculation, we obtain

Thanks to the global bifurcation theorem, the local bifurcation curve can be extended to a smooth global curve , and

Moreover, according to Theorem 6.4.3 in document [22], the global curve must satisfy one of the following three alternatives:

  is unbounded in   There exists a constant such that   There exists a function such that

where is the complementary space of and satisfies

The positive one can be defined as follows:

Next, we prove . Applying reduction to absurdity, we assume that . There exists a sequence

such that

in , where , and is a nonnegative solution of system (1) corresponding to . According to the strong maximum principle [22, 24], we know that satisfies one of the three following alternatives:

(i)(ii)(iii)

Similar to the method to Theorem 1 in [24], we can find a contradiction to each of these cases. Therefore, we have

Since equation (60) holds, it is obviously impossible for case (b) to occur. Because in ; obviously, case (c) does not impossible hold. Therefore, the global curve just belongs to case (a), combined with Lemma 3.2; the projection of the global curve on the axis contains .
The specific analysis and calculation of the direction of local bifurcation branches have been given in the previous section, so we have completed the proof of this theorem.

4. Conclusion

In this work, the effect of spatial degradation on the steady-state problem of a predator-prey system with B-D functional response has been investigated. By studying the bifurcation structure of the system, sufficient conditions for the coexistence of two species are obtained. However, due to the phenomenon of space degradation, there are some phenomena that are different from the traditional predator-prey system (i.e., all the coefficients of the system are normal or positive). Specifically, for the traditional predator-prey system, the two species do not coexist when the predator’s own growth rate is large, but for the system (1), the growth rate of the predator is very large, and the two species can still coexist in the common habitat , even if the predator growth rate is very large. The results show that spatial degradation has a significant effect on the steady-state behavior of the system. On the other hand, our results are essentially different from the results of the Lotka–Volterra predator-prey system (). Specifically, under the case of the strong growth rate on the prey, for any large , the two species of the system (1) can coexist, but for the smaller , the two species cannot coexist. However, for the Lotka–Volterra predator-prey system, no matter how small the predator’s self-growth rate , the two species can continue to coexist, but for the larger , the two species cannot coexist. The result reveals that the B-D reaction function also has a significant effect on the steady-state behavior of the system.

Data Availability

This article belongs to the qualitative analysis of the dynamic system and no data were involved.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

All authors participated in every phase of research conducted for this paper. All authors read and approved the final manuscript.

Acknowledgments

The work was partially supported by the National Natural Science Youth Fund of China (12001425), the Natural Science Basic Research Plan in Shaanxi Province of China (No. 2020JM-569), the Shaanxi Province Department of Education Fund (18JK0393), the Project of Improving Public Scientific Quality in Shaanxi Province (No. 2020PSL(Y)073), and Test & Measuring Academy of Norinco, Huayin, Shaanxi Province.

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