Permanence of Diffusive Models for Three Competing Species in Heterogeneous Environments

Abstract

We address the question of the long-term coexistence of three competing species whose dynamics are governed by the partial differential equations. We obtain criteria for permanent coexistence in a Lotka-Volterra system modeling the interaction of three competing species in a bounded habitat whose exterior is lethal to each species. It is also proved that if the intercompeting strength is very weak, the system is always permanent, provided that each single one of the three species can survive in the absence of the two other species.

1. Introduction

Reaction-diffusion systems are some of the most widely used models for population dynamics in situations where spatial dispersal plays a significant role. Generally speaking, the dynamical behaviors of the systems with two competing species are relatively simple and have been widely investigated in the past years. However, the investigation and knowledge about the systems with three competing species are very limited; see [1–5] for some related results. It is well known, in general, that the dynamical behavior of systems with three competing species may be extremely complex, even in nondiffusive case, namely, ODE case; see, for example, [6–9] and the references therein.

The object of this paper is to study the problem of coexistence for three competing species dispersing through a spatially heterogeneous region. We model the population dynamics of the species with a system of three diffusive Lotka-Volterra equations.

The model we consider has the general form with on , where is a bounded domain and , , are all continuous in . The variables represent population densities of the competing species. The boundary condition describes a situation that the boundary of is lethal to the species.

We will use the criterion of permanence to characterize coexistence. A system is said to be permanent if any solutions with all components positive initially must ultimately enter and remain within a fixed set of positive states that are strictly bounded away from zero in each component. For some investigations of coexistence characterized by permanence, mostly in the case for two interacting species, see [10–12], and also some parts of the books [13, 14].

This paper is organized as follows. In Section 2, we introduce some necessary material about semiflow and state V. Hutson’s average Lyapunov theorem which will be the basic tool in the proof of our main theorem. We also discuss the dissipative property of our systems, which is necessary for using V. Hutson’s average Lyapunov function theorem. In Section 3, we establish our main permanent result, Theorem 11. We also prove that if the intercompeting strength is “very weak,” the system (1) is always permanent, provided that either single one of the three species can survive with the absence of the two other species. In Section 4, we give some summery and discussion of the results of this paper.

2. Semiflows and Dissipativity

In this section, we will make some preparations for establishing our main results in the next section. Firstly, we will introduce some terminologies and results about semiflows (semigroups) for readers’ convenience. The materials can be found in [15] or [13, 14].

Let be a metric space, with points in being denoted by and subsets by . The following two unsymmetric distances of sets will be used: The triple is said to be a semiflow (or semigroup), if is continuous and satisfies:(i),(ii),

for all . For convenience, we often write . The symbols and denote the semiorbit through and the omega limit set of , respectively, and the equivalent expressions for sets are defined by taking unions.

A solution through is a continuous map such that and for . The range of is denoted by and is called an orbit through .

A set is said to be forward invariant if and invariant if . The semiflows is said to be dissipative if there is a bounded set such that for all . is said to be a global attractor of the semiflow if it is compact invariant and for all bounded , , where .

Theorem 1. Let be complete and suppose that the semiflow is dissipative. Assume that there is a such that is compact for ; then there is a nonempty global attractor.

Consider next the concept of permanence in the abstract semiflow context. We suppose that , where is open, and assume that , are forward invariant. In relation to the remarks in the introduction, will consist of the states with at least one species identically zero.

Definition 2. The semiflow is said to be permanent if there exists a bounded set with such that for all .

A set is said to be strongly bounded if it is bounded and . is said to be a global attractor relative to strongly bounded sets if it is a compact invariant subset of and for all strongly bounded .

Permanence is obviously an asymptotic property. It can thus be studied by examining the semiflow restricted to a forward invariant set derived from an -neighbourhood of the global attractor of Theorem 1. Set then , the closure of , and take . The following Hutson’s theorem on average Lyapunov functions is the basic tool for establishing our main theorem in Section 3.

Theorem 3 (see [15]). Assume that the conditions of Theorem 1 hold, and let , be as defined above. Suppose that is continuous, strictly positive, and bounded, and for define Then the semiflow is permanent if

For our application of the last theorem, we will cast the system (1) to the abstract frame of the semiflow. It is well known that if (1) is simply viewed as a parabolic system, it actually generated a semiflow in , and solutions which belong to must have Hölder-continuous second derivatives on (see [13]). Thus, we may use maximum principles to obtain a priori bounds, even though we will ultimately want to view our semiflow as acting on . Then by maximum principle (or comparison theorem), any solution of (1) must satisfy a uniform bound of the form for some constants , , after finite time. So the semiflow generated by (1) is dissipative. We state it as follows.

Theorem 4. The semiflow generated by the system (1) is dissipative.

3. Permanent Coexistence Results

In this section, we establish criteria of permanent coexistence of the system (1). The main tool is Hutson’s average Lyapunov function theorem stated in Section 2, and the key step is to set up a suitable average Lyapunov function. To construct the average Lyapunov functions, we must have a detailed knowledge of the -limit set of the generated semiflow in the boundary of the positive cone.

The maximum principle implies that solutions of (1) with nonnegative nonzero initial data for a given component will have that component strictly positive in for all . In the case of Dirichlet conditions, such solutions will have normal derivatives on which are bounded above by a negative constant. Hence, the only trajectories which remain in the boundary of the positive cone have one or both components identically zero. Thus, to determine the -limit set of the semiflow generated by (1) on the boundary, we need only consider the steady state solutions (equilibrium points) of subsystems of (1).

Let denote the principal eigenvalue for problem

In the rest of this paper, we denote

The point is always a steady state, which means that there are no species in the domain. With the absence of two species and only the other one species left, system (1) becomes a scalar equation, and we have the following well-known result (see [14]).

Lemma 5. Suppose that for . Then the following problem for , has a unique positive steady state solution which is globally approximately stable.

Remark 6. Lemma 5 tells us that if for , , then either single one of the three species can survive in the absence of the other two species.

Now we consider the subsystem of (1) with only one species to be absent. There are three cases.

Case 1. One has ,

Case 2. One has ,

Case 3. One has ,
With some modification of lower-supper solution methods of parabolic systems (see [14]), it is easy to prove the following results.

Theorem 7. With , as defined in (6), (1)if  the system (8) has a steady state solution with for ; similarly,  (2)if  the system (9) has a steady state solution with , for ; (3)if  the system (10) has a steady state solution with , for .

Proof. We only prove the part (1); the proof of part (2) and part (3) are similar.
Let be a positive eigenfunction of the principal eigenvalue for the eigenvalue problem (5). Choose sufficiently small, for all . So, is a set of upper and lower solutions for in (8).
Similarly, choose sufficiently small; is a set of upper and lower solutions for in (8).
By coupled upper and lower theorem (see [16, Theorem 1.4-2]), the system (8) has a steady state solution with , for .

Remark 8. By the comparison principle, it is easy to see that, in , we have , , , , and , .

Now, we consider the unique problem of the steady state solutions above.

Let denote the unique positive solution of for any . For brevity, we denote They are all positive functions in .

Theorem 9. Assume that all the hypotheses of Theorem 7 are satisfied.(1) If  then the strictly positive steady state solution of system (8) is unique; similarly (2)if  then the steady state solution of system (9) is unique; (3) if  then the steady state solution of system (10) is unique.

Remark 10. For fixed functions , , , hypothesis (17) will be satisfied for , sufficiently small. This is true because of Hopf’s strong maximal value theorem, and also because (resp., ) increases as (resp., ) decreases for . Thus (resp., ) decreases as (resp., ) decreases. Similarly, hypotheses (18) and (19) will be satisfied if , , , and are sufficiently small.

Proof. We only give the proof of part (1) in Theorem 9, since the arguments of part (2) and part (3) are similar.
Assume that , are two strictly positive steady state solutions of system (8) in .
Let then
Since is a strictly positive solution of with , the number must be the smallest eigenvalue of the above problem. Moreover, by variational properties, we have for any which vanishes on . Similarly, since is strictly positive solution of with , the number must be the smallest eigenvalue of the above problem. Moreover, for any which vanishes on . Multiplying the first equation of (21) by , the second by , integrating over , and adding, we deduce from (23) and (25) that By comparison of scalar equations using upper and lower solutions we can readily obtain for , From (27), we have in . It follows from (17) that Then it is easy to see that the quadratic expression in the integrand of (26) is positive definite for each . Consequently, we must have and identically equal to zero in . That is in .

Now we are ready to develop the permanent coexistence results of the system (1). To construct the average Lyapunov functions, we will use the positive eigenfunctions , , and corresponding to the principal eigenvalues , , and of respectively.

Theorem 11. Suppose that (11)–(13), (17)–(19) hold, and the principal eigenvalues , , and of (30), (31), and (32), respectively, are all positive. Assume also that for all ; then the semiflow on generated by (1) under homogeneous Dirichlet boundary conditions is permanent.

Proof. The hypotheses imply that (1) is dissipative and that the -limit set in the boundary of the positive cone consists of the following seven cases: , , , , , , and . Thus we need only find a suitable average Lyapunov function to conclude permanence. Since the semiflow is dissipative, we may restrict our attention to a bounded absorbing subset of the positive cone in that contains the global attractor whose existence is asserted in Theorem 1. Let denote the intersection of with the boundary of the positive cone. Choose eigenfunctions , , for (30), (31), and (32), respectively, and define We have for that we need for and for . Let . Computation using (34) yields
We have for
Similarly,
Since is bounded, are bounded below on , so that the ratios are bounded below. It follows that has a strictly positive lower bounded since the expression inside the exponential in (36) is bounded away from . To see what happens as , we must examine how behaves as , . By the continuity of the semiflow , , , and will be uniformly close to , , and , respectively, for sufficiently small. Hence, if we can show that the expression always has a positive as , then by (35)–(39), we have for and sufficiently small.
If we let , then
Then As , Also so that as , Since , the first ratio of integrals in (42) has limit as . For the latter two ratios of integrals, note that as .
By the assumption (33), we have Hence we have As , computations similar to (45) and (46) show that
By assumption (33), we have
Hence we have As , computations similar to (45) and (46) show that
By assumption (33), we get Hence we have As , Also so that as , For the second ratio of integrals, Also so that as , For the third ratio of integrals, as , Hence we have Using the same method, we can get the conclusion about the other two cases.
As , and as , In conclusion so that for and sufficiently small, and permanence follows from the abstract results of the previous section.

Now we are ready to prove an interesting result, namely, permanent existence always holds among three “very weak” competing species. The result may be known, but to our knowledge, it does not appear in the published literature.

Theorem 12. Suppose that , for , . Then there is a constant , such that if for , with , the semiflow on generated by (1) under homogeneous Dirichlet boundary conditions is permanent.

Proof. For fixed and in , , when (, , ) are sufficiently small, conditions (11)–(13) and (17)–(19) are satisfied, and by the variational expression of principal eigenvalues (see [13]) and the boundedness of (see Remark 8), the principal eigenvalues , , and of (30), (31), and (32), respectively, are all positive. And then it is easy to see that condition (33) is satisfied when () are even smaller, if necessary. The proof is completed by Theorem 11.

4. Summary and Discussion

The important aspect of the approach of our main result, Theorem 11, is that it produces conditions that in terms of coefficients describing the strength of interactions. The criterions depend only on the behavior near the species equilibria of the two-dimensional subsystems, which is relatively easy to examine experimentally. The eigenvalues depending on the coefficients can be computed explicitly in simple cases, and there is a huge literature available on their estimation, numerical approximation, qualitative properties, and so on. Eigenvalue-based criteria for coexistence can then be used to reach biological conclusions that have been difficult to deduce through other methods.

Theorem 12 tell us that if the intercompeting coefficients are sufficiently small, the system (1) is always permanent, provided that in , which is, by Lemma 5, every single species can survive when the other two species become extinct. This sounds reasonable, and we guess that this is true for systems of any finite multiple competing species. A related topic on the systems of finite multiple competing species with constant coefficients and Neumann boundary condition can be found in [17], Theorem 9.

The significant feature of our conditions for permanence is that they do not require any special assumptions the uniqueness of coexistence states, the existence of a global Lyapunov function, or other such properties, and so the methods used here can be used to other more general situations.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the anonymous reviewers for their helpful comments and suggestions. This work was supported by Grant 13ZR1430100 from the Natural Science Foundation of Shanghai, China.