SPECIES SURVIVAL VERSUS EIGENVALUES

The purpose of this work is to analyze mathematical models describing the behavior of individuals inhabiting an environment with spatial heterogeneity. The latter means that the environment influences the reproductivity and mobility of the species. This effect is obtained if we model the density function of these organisms by evolution partial differential equations with nonconstant coefficients. More precisely, we introduce a function which corrects the reproductive capabilities of these species and characterizes the most and least favorable regions for its development. With this at hand, we wish to know if such an environment constitutes a refuge, that is, if this environment has good enough conditions for the studied species persistence, which in mathematical terms implies the existence of a stable nontrivial stationary solution—the limit density of these individuals in the steady state. Such a stationary solution is obtained by making use of the fibering method [7, 13, 14, 15, 16], introduced and developed by Pohozaev. This method implies that the fundamental eigenvalue of a certain spectral problem must be of a determined magnitude. This paper is organized as follows. In the next section, we present the details of the considered mathematical model. In Section 3, we apply the fibering method to a problem studied by Cantrell and Cosner [3, 4, 5, 6] and discuss the relationship between their approach and the one used here. Section 4 contains concluding remarks and comments. In the appendix, we adapt the method to a particular case considered by Ludwig et al. [10].


Introduction
The purpose of this work is to analyze mathematical models describing the behavior of individuals inhabiting an environment with spatial heterogeneity.The latter means that the environment influences the reproductivity and mobility of the species.This effect is obtained if we model the density function of these organisms by evolution partial differential equations with nonconstant coefficients.More precisely, we introduce a function which corrects the reproductive capabilities of these species and characterizes the most and least favorable regions for its development.With this at hand, we wish to know if such an environment constitutes a refuge, that is, if this environment has good enough conditions for the studied species persistence, which in mathematical terms implies the existence of a stable nontrivial stationary solution-the limit density of these individuals in the steady state.
Such a stationary solution is obtained by making use of the fibering method [7,13,14,15,16], introduced and developed by Pohozaev.This method implies that the fundamental eigenvalue of a certain spectral problem must be of a determined magnitude.
This paper is organized as follows.In the next section, we present the details of the considered mathematical model.In Section 3, we apply the fibering method to a problem studied by Cantrell and Cosner [3,4,5,6] and discuss the relationship between their approach and the one used here.Section 4 contains concluding remarks and comments.In the appendix, we adapt the method to a particular case considered by Ludwig et al. [10].

The mathematical model
We consider a species which inhabits a limited plane region Ω ⊂ R 2 .We suppose that Ω is a domain, that is, an open and connected set.Further, we admit that the population dynamics is of reaction-diffusion type and movement and reproduction capacities depend on the position in Ω.More precisely, let u = u(t,x, y) be a positive function which is the density of the individuals of the population at the moment t ≥ 0 near the point with coordinates (x, y), then the initial/boundary value problem governing the density u is: Here, the positive and bounded function D denotes the diffusion coefficient, which is permitted to depend on the spatial variables (x, y).It is assumed that there exist positive constants c 1 and c 2 such that 0 < c 1 ≤ D(x, y) ≤ c 2 .The function γ, which measures the interference of the environment in the reproductive capacities of the studied population u, may also depend on (x, y).We suppose that γ is a bounded function such that |γ| ≤ 1.Moreover, we admit that γ may assume negative values elsewhere, indicating in this way the existence of some absolutely improper subregions for the development of the population.However we will consider the environmental saturation as a constant K in Ω.This constant is known as the support capacity for the subregion where γ > 0. As in the classical Verhulst model, the constant r > 0 is the rate of intrinsic growth.Finally, the bounded and nonnegative function u 0 is the initial population distribution.The analysis of species survival via models like the problem (2.1) is based on the existence and stability of stationary solutions, that is, solutions of the following problem: The existence or nonexistence of nontrivial solutions to the above stationary problem determines the chances for success in the colonization of the environment.In (2.2), we scale the independent variables: ) ) we also use the dimensionless unknown function Introducing the new functions where |Ω| denotes the area of Ω, we obtain the following problem to be analyzed: Above, U denotes the image of Ω via the transformation (2.3), (2.4) with |U| = 1 (for notation simplicity we have omitted the bars).
We also introduce a control parameter α > 0 in order to modulate the reproductivity differences in the environment quality without changing its saturation and sign.Therefore, we will study the following variant of our problem (2.8): (2.9) Since (2.9) is of divergence form without first-order terms, we will use variational techniques.
To begin with, let the Sobolev space W := H 1 0 (Ω) be considered with the norm This norm is equivalent to the usual norm of H 1 0 (U).Indeed, this follows from the assumptions on D and from the Poincaré inequality.Then, u ∈ W is called a weak solution of (2.9) if for all v ∈ W, (2.11) The solutions of the variational problem will be identified with the critical points of the following Euler functional: where Biological models in the terms of reaction-diffusion equations are well known for a long time (see for instance [9,11,12]).The present model was proposed by Cantrell and Cosner [2,3,4,6] (see also [8]).However in the cited works, these authors emphasize its importance for the study of species persistence in heterogeneous environments.Moreover, they admit negative values of some of the parameters.Motivated by these papers, we obtain here similar results using other methods, namely, the fibering method of S. I. Pohozaev.This will be done in the next section.

The fibering method
In the study of the stationary problem (2.9), we will use the fibering method introduced and developed by Pohozaev in [13,14,15,16].This method provides a powerful tool for proving existence theorems, in particular for problems which obey certain kinds of homogeneity.In [7], Drábek and Pohozaev have applied the method to an equation involving the p-Laplacian operator.We will describe here an adapted version of the fibering method to our specific problem.The exposition will follow [7,16] closely.We present enough details of the results in order to compare with those obtained in [2,3,4,6] and discuss the relationship between them, in particular their importance to biology.
To begin with, we consider the Euler functional defined in (2.12).Clearly, α (u) ∈ C 1 .Critical points of α (u) are then weak solutions of the problem.Later we will associate to α (u) another functional with additional properties.For this purpose, the magnitude α will be compared with the fundamental eigenvalue λ 1 of the problem for any ψ ∈ W. We recall that the fundamental eigenvalue λ 1 of the problem (3.1) can be characterized as follows: where λ 1 is simple and positive (see [1] and the references therein).Further, following [13,14,15,16], for u ∈ W, we set where t = 0 is a real number and v ∈ W (since we look for nontrivial solutions, the assumption t = 0 is natural).Substituting (3.3) into (2.12),we obtain We choose as fibering functional the principal part of α , that is, In our specific case, (3.6) assumes the form Luiz Antonio Ribeiro de Santana et al. 119 We obtain Obviously, the following conditions are necessary for (3.8) to make sense: Substituting (3.8) in (2.12), we obtain the induced functional The induced functional α obeys the following properties: (1) for any that is, the functional α is homogeneous of degree 0. Moreover as in [7], one can assume that the critical points of α are nonnegative.The next two properties are direct consequences of the general fibering method described in [13,14,15]; (3) let v ∈ W be a critical point of α such that F(v) = 0 and H α (v)/F(v) > 0, then the function where t > 0 is determined by (3.8), and is a critical point of α ; (4) we consider a constraint where then every critical point of α with the constraint Our first aim is to prove the existence of a critical point of α with an appropriate condition Ᏺ(v) = c which in turn will be an actual critical point of α and hence a critical point of the Euler functional α -the weak solution of (2.9).See [7,16] for further details.
Applying the described method to the problem (2.9), one can obtain the existence of solutions whose multiplicity depends on the magnitude of the fundamental eigenvalue λ 1 .There are two cases to be considered: (1) α ≤ λ 1 ; (2) α > λ 1 .In the first case, we get the existence of a positive nontrivial solution of (2.9) choosing the constraint which satisfies the nondegeneracy condition since if However, F(v) must be positive for (3.8).Hence, the fibering method cannot be applied immediately in this case.This is compatible with the results of Cantrell and Cosner [3] which predict the nonexistence of positive solution if α ≤ λ 1 .
With regard to the case α > λ 1 , the fibering method should give, in principle, two critical points for the functional (2.12) which are positive functions.However, in order to obtain one of these, we need the positivity of the functional F. Therefore, we cannot get it in this case.Why this occurs is commented in the next section.Now, we will obtain the "other" positive solution of (2.9) with α > λ 1 .Let be a constraint.The nondegeneracy condition is satisfied since Then the induced functional α becomes With t determined by (3.8), we look for a critical point v of α such that For this purpose, we look for a function which attains the minimum m α of the problem First, we must prove that

.22)
Let e 1 be the positive eigenfunction associated to the fundamental eigenvalue (it satisfies (3.1) for any ψ ∈ W-see [1]).Then and hence it is easy to see that we can find a constant t 1 such that F(t 1 e 1 ) = −1.Therefore W − is not empty.Moreover, by the variational characterization of λ 1 , which implies that the infimum m α is negative.We are going to prove the existence of a positive solution to (2.9) for α ∈ (λ 1 ,λ 1 + L), where L > 0 is a constant by a contradiction argument.Suppose that this is not true, and that there exists a sequence ε k → 0 such that for any α k := λ 1 + ε k , the minimization problem (3.21) does not have a solution.For any integer k, let (v k n ) ∞ n=1 be a minimizing sequence for the problem (3.21), that is, n=1 would be bounded for an integer number k, we may assume without loss of generality that it converges weakly in W to some v k when n → ∞.By the weak continuity of the functionals F and G and the weak lower semicontinuity of the principal part H α , one can deduce, letting n → ∞, that By the definition of m α k in (3.21), the opposite inequality holds, which is a contradiction.Thus we may consider (v k n ) ∞ n=1 to be unbounded: where Since w k n = 1, we may assume that w k n converges weakly in W to a function w k ∈ W as n → ∞.By w k ≤ 1, passing to a subsequence of w k converging weakly to w ∈ W when k → ∞, we have w ≤ 1. (3.30) By the definitions of v k n and w k n , we get the inequality Hence and by (3.30), letting n → ∞ and k → ∞, we obtain However the variational nature of λ 1 (see the Rayleigh quotient (3.2)) implies that the opposite inequality holds.Therefore, we can find a constant k 1 ∈ R such that (3.33) By the 3-homogeneity of the functional F we conclude that Letting n → ∞ and k → ∞, we obtain by the weak continuity of F F(w) = 0, (3.35) and, by (3.33), which contradicts (3.23).Therefore, with α = 1, we obtain a solution u of the problem (2.9) if λ < 1.Indeed, let v be the minimizer of the problem (3.21).Introduce u by where t(v) is defined in (3.8).Then by the properties (3) and ( 4) of the functional α , u is a weak solution of (2.9).The references in [7,16] ensure the positivity and differentiability of the solution u.In this way we have proved the following theorem.
Theorem 3.1.The boundary value problem (2.9) has at least one positive solution

Comments and concluding remarks
In [7], Drábek and Pohozaev investigated the existence of positive solutions of the following quasilinear problem: Here λ, p, q are real numbers, a(x), b(x) are given functions of x ∈ R n , and ∆ p is the p-Laplace operator which for p = 2 coincides with the usual Laplacian.The p-Laplacian for p = 2 is much less important for biological modelling (if any) than the classical p = 2 which embodies the ubiquitous Fickian/random diffusion process.For this reason, we let p = 2 and obtain the semilinear boundary value problem We recall some of the assumptions on the parameters used in [7].The functions a(x) and b(x) are supposed to be bounded in Ω; ) where u 1 (x) is the first positive eigenfunction of the p-Laplace operator.For the Cantrell-Cosner problem (2.9), we have α = λ, a = γ, b = −1, and s = 3. Obviously, the conditions ( are fulfilled, but the condition (4.5) is not satisfied since That is the reason why it was not possible to get a second solution in the previous section.However, if we consider the model (4.2) with (4.3), (4.4), (4.5), and (4.6), which is in the spirit of the already cited works of Cantrell and Cosner, it is clear that a straightforward application of the fibering method will give at least two different positive solutions, a result which begs for further biological interpretations.

Appendix Application of the fibering method to a simple boundary value problem
In this section, we apply the fibering method to a simple boundary value problem in dimension one.This example serves as an elementary introduction to the essential points of the method, which is a powerful tool to be used in more complicated situations.The fibering method, due to S. I. Pohozaev, gives us information about existence and multiplicity of nonnegative and nontrivial solutions to several types of boundary value problems, where the partial differential equation is generally non linear.The presence of nonlinearity comes from problems of biological nature: for instance, in the study of generation of spatial patterns and persistence of species in regions explored by these species [11].
The introductory example chosen concerns the existence and multiplicity of solutions to the stationary Fisher/KPP equation under Dirichlet boundary condition: The problem (A.1) has been studied by Ludwig et al. [10] using the first integral method.This model, which is appropriately dimensionless, serves to describe the steady state distribution of individuals which spread out randomly in a one-dimensional medium with a lethal boundary (extremal points of the interval (0,1)) condition of the Dirichlet type.
It is also assumed that reproduction of these species obeys the Verhulst model of logistic growth.
In order to use the fibering method, we need to reformulate the problem (A.1) in its equivalent variational form, that is, we need to consider the function which solves (A.1) by minimizing the following Euler functional: where which are both weakly continuous functionals.There exists an equivalence among weak solutions of (A.1) and critical points of (A.2), for when we compute the Gâteaux derivative of (A.2), we have Hence, the expression inside the parentheses in the last integral must vanish if and only if u solves the differential equation in problem (A.1).We naturally choose as the function space the Sobolev space H 1 0 (0,1) which consists of functions belonging to L 2 (0,1) having a generalized derivative also in L 2 (0,1) and vanishing for x = 0 and x = 1.We use for simplicity the following norm: which is equivalent to the usual norm in H 1 0 (0,1) by Poincaré's inequality.It is useful to note here that, by the embedding W 2 1 (0,1) C 0,σ (0,1) for some positive σ, the solution of (A.1) is a continuous function.
(i) Considering any 0 = t ∈ R, we have which implies the 0-homogeneity of this induced functional.
(ii) As a corollary of the above remark (i), the Gâteaux derivative of f (v) in the direction v is equal to zero because From the above construction the following property holds: This result comes from the definitions of H (v) and F(v) (see (A.3) and (A.4)).Therefore, if v c is a critical point of f (v), then |v c | is also a critical point of the same functional and we can consider the critical point(s) of the induced functional f (v) as positive functions in (0,1).
(iv) The critical points of the induced functional f (v) are also critical points of the same functional but with a specific type of constraint, that is, if v is a critical point of f (v) then, it is also a critical point of f (v) subject to the fibering constraint where H(v) is any differentiable functional satisfying which is named the nondegeneracy condition.
The existence and multiplicity of solutions of the induced functional f (v) is intimately related to the magnitude of the constant .We should consider the following two cases: Describing the first case with brevity, the fibering method indicates the existence of at least one positive weak solution of problem (A.1) under the restriction which satisfies the nondegeneracy condition in this case, because We should have, by (A.12) and (A.18), the positivity of the functional F(v).But this is in contradiction to its own definition (cf.(A.4)).The method does not give us information about this case, which was expected since it is known that there are no positive solutions in this case [10].According to Ludwig et al. [10], the interval (0,1) is not a refuge in this case. [