Permanence in N species nonautonomous competitive reaction – diffusion – advection system of Kolmogorov type in heterogeneous environment

One of the important concept in population dynamics is finding conditions under which the population can coexist. Mathematically formulation of this problem we call permanence or uniform persistence. In this paper we consider N species nonautonomous competitive reaction–diffusion–advection system of Kolmogorov type in heterogeneous environment. Applying Ahmad and Lazer’s definitions of lower and upper averages of a function and using the suband supersolution methods for PDEs we give sufficient conditions for permanence in such models. We give also a lower estimation on the numbers δi which appear in the definition of permanence in form of parameters of system { ∂ui ∂t = ∇[μi∇ui − αiui∇ f̃i(x)] + fi(t, x, u1, . . . , uN)ui, t > 0, x ∈ Ω, i = 1, . . . , N, Diui = 0, t > 0, x ∈ ∂Ω, i = 1, . . . , N.


Introduction
A main problem in population dynamics is the long-term development of population.Uniform persistence (sometimes also called permanence), coexistence and extinction describe important special types of asymptotic behavior of the solutions of associated model equations.In this paper we consider the N species nonautonomous competitive reaction-diffusionadvection system of Kolmogorov type x, u 1 , . . ., u N )u i , t > 0, x ∈ Ω, i = 1, . . ., N, (1.1)  J. Balbus which is endowed in appropriate boundary conditions.In the context of ecology u i (t, x) denote the densities of the i-th species at time t and a spatial location x ∈ Ω, Ω ⊂ R n is a bounded habitat and fi (x) = lim inf t−s→∞ 1 t − s t s f i (τ, x, 0, . . ., 0)dτ, (i = 1, . . ., N) accounts for the local growth rate.If the environment is spatially heterogeneous i.e., fi (x) is not a constant then the population may have tendency to move along the gradient of the fi (x) (i = 1, . . ., N) in addition to random dispersal.The constants α i for 1 ≤ i ≤ N measures the rate at which the population moves up the gradient of fi (x).Through this paper we only consider the case α i ≥ 0, for 1 ≤ i ≤ N i.e., the populations move up in the direction along which fi is increasing.Models of ecology are described by ordinary differential equations (see e.g.[11,12,26,27,30]) or partial differential equations (see e.g.[3,4,6,10,18,19,21,24,29,31]).In the case of autonomous ODE sufficient conditions for permanence are given in a form of inequalities involving an interaction coefficients of the system (see e.g.[1]).
In [2] S. Ahmad and A. C. Lazer considered an N species nonautonomous competitive Lotka-Volterra system.The authors introduced a notion of upper and lower averages of a function.They found sufficient conditions which guarantee that such system is permanent and globally attractive.
In [25] we extended their results on N species nonautonomous competitive system of Kolmogorov type.
The models of ODEs do not take into account spatial heterogeneity.They give the temporal changes in terms of the global population while partial differential equations give the temporal changes at each point in space in terms of the local densities and the spatial gradients.Dispersal of individuals has important effects from an ecological point of view and in the biological literature we can find that temporally constant tends to reduce dispersal rates (see e.g.[9]) or temporal changes in the environment tends to lead to higher dispersal rates (see e.g.[16]).
One of the popular models which take into account spatial heterogeneity is reactiondiffusion system of PDE The system (1.2) is an example of model of the population growth with unconditional dispersal.Unconditional dispersal does not depend on habitat quality.This type of dispersal is investigated by many authors, see for example [3,10,14,15,17,22,23].In [23] the authors investigated uniform persistence for nonautonomous and randomly parabolic Kolmogorov systems via the skew-product semiflows approach.They obtained sufficient conditions for uniform persistence in such systems in terms of Lyapunov exponents.
In [3] we studied N species nonautonomous reaction-diffusion Kolmogorov system with different boundary conditions, either Dirichlet or Neumann or Robin boundary conditions.We gave sufficient conditions for permanence in such system.Those conditions are given in a form of inequalities involving time averages of intrinsic growth rates, interaction coefficients, migration rates and principal eigenvalues.In nature species do not move completely randomly.Their movements are a combination of both random and biased ones.Such models are called models with conditional dispersal.The most popular model which takes into account some amounts of random motion and a purely directed movement dispersal strategy is reaction-diffusion-advection system.This type strategy is considered widely in literature (see e.g.[7,8,10,18]).
The logistic reaction-diffusion-advection model for the population growth has the following form (1. 3) The constant α measures the rate at which the population moves up the gradient of m(x).In [8] the authors examined the case α ≥ 0. The boundary conditions ensures that the boundary acts as a reflecting barrier to the population i.e., no-flux across the boundary.Belgacem and Cosner [4] studied (1.3) with both no-flux and Dirichlet boundary conditions.The authors showed that the effects of the advection term αu∇m depend critically on boundary conditions.However, for no-flux boundary condition sufficiently rapid movement in the direction of m(x) is always beneficial.In the case of Dirichlet boundary condition movement up the gradient of m(x) may be either beneficial or harmful to the population.The authors studied the effect of drift on the principal eigenvalues of certain elliptic operators.The eigenvalues determine whether a given model predicts persistence or extinction for the population.
In [8] Cosner and Lou showed that the effects of advection depend crucially on the shape of the habitat of the population.In the case of convex habitat the movement in the direction of the gradient of the growth rate is always beneficial to the population.In the case of nonconvex habitat such advection could be harmful to the population.
In [7] Chen et al. investigated a two species model of reaction-diffusion-advection in Ω × (0, ∞) with no-flux boundary conditions They assumed that both species have the same per capita growth rates denoted by m(x).
In biological point of view it may mean that the two species are competing for the same resources.They assumed also that m(x) is a nonconstant function.The resource is usually spatially unevenly distributed.Because of that the movement of species is purely random.The model (1.4) consist of two component: random diffusion (µ∇u and ν∇v) and directed movement upward along the gradient of m(x) (α(∇m)u and β(∇m)v).The authors showed that if only one species has a strong tendency to move upward the environmental gradients the two species can coexist since one species mainly pursues resources at places of locally most favorable environments while the other relies on resources from other parts of the habitat.However, if both species have such strong biased movements it can lead to overcrowding of the whole population at places of locally most favorable environments which causes the extinction of the species with stronger biased movements.
In this paper we find sufficient conditions for uniform persistence in the N species nonautonomous competitive system of reaction-diffusion-advection.In contrast to [7,13,20,21] we assume that all species have a different intrinsic per capita growth rates, and we take into account the influence of the jth species of the growth rate of the ith species.The investigation of nonautonomous systems is of great importance biologically since in nature, many systems are subject to certain time dependence which may be neither periodic nor almost periodic.This paper is organized as follows.
In Section 2 we introduce basic assumptions and some results about the principal eigenvalue of the eigenproblem (2.1).We also formulate auxiliary results on the behavior of the positive solutions.
In Section 3 we state and prove the main theorem of this paper.We formulate average conditions which guarantee that system (ARD) is permanent.
In Section 4 we formulate the stronger inequalities which give a lower bound on the population densities in term of interaction coefficients of system (ARD).

Preliminaries
Consider a nonautonomous competitive N species model of reaction-diffusion-advection where fi (x) = lim inf t−s→∞ 1 t−s t s f i (τ, x, 0, . . ., 0)dτ are nonconstant functions for i = 1, . . ., N. Ω ⊂ R n is a bounded domain with the sufficiently smooth boundary ∂Ω, µ i > 0 is a diffusion rate of the i-th species, α i ≥ 0 measure the rate at which the population moves up the gradient of the growth rate fi (x) of the i-th species and f i (t, x, u 1 , . . ., u N ) is the local per capita growth rate of the i-th species.
We define the operator Further we define the boundary operator D i which is either the Dirichlet operator or the operator Denote by λ i (α i ) the principal eigenvalue of the eigenproblem In the case of Dirichlet boundary conditions it is known that (2.1) will always have a unique positive principal eigenvalue λ 1 i (α i ) which is characterized by having a positive eigenfunction.In the case of no-flux boundary conditions we need the following lemma.Lemma 2.1 (see [4]).The problem (2.1) subject to no-flux boundary conditions has a unique positive principal eigenvalue α i (α i ) characterized by having a positive eigenfunction if and only if Definition 2.2.System (ARD) is permanent if there are positive constants δ i , δ i such that for each positive solution u(t, x) = (u 1 (t, x), . . ., u N (t, x)) of (ARD) where the limit is uniform in x ∈ Ω.
We introduce now a first assumption for a functions f i which guarantee the existence and the uniqueness of local classical solutions to an initial value problem for (ARD).
For each 1 ≤ i ≤ N there holds Proof.Fix a positive solution u(t, x) = (u 1 (t, x), . . ., u N (t, x)) of (ARD).Denote by v i (t, x), 1 ≤ i ≤ N, t ≥ 0, x ∈ Ω, the solution of the following boundary value problem for all t > 0, x ∈ Ω.For T ∈ (0, τ max ) and 1 ≤ i ≤ N put We prove that for t ∈ [0, T] and x ∈ Ω.We have In the case of Dirichlet boundary conditions we have for t > 0, x ∈ Ω and i = 1, . . ., N. In the case of no-flux boundary conditions we have Again we have (2.5).In a similar way we show that for t ∈ (0, T], x ∈ Ω, i = 1, . . ., N. By (2.4), (2.5) and (2.6) we have the desired inequality.
For i = 1, . . ., N, the function f i (t, x, 0, . . ., 0) is called the intrinsic growth rate of the ith species.In [7] the authors assume that the two species have the same per capita growth rate.We assume that all species have a different per capita growth rates.For this reason, our model is more realistic.To reflect the heterogeneity of environment, we assume that fi (x), i = 1, . . ., N are non constant functions.The functions fi (x) can reflect the quality and quantity of resources available at the location x, where the favorable region {x ∈ Ω : fi (x) > 0} acts as a resource and the unfavorable part {x ∈ Ω : fi (x) < 0} is a sink region.
The assumption below is a standard boundedness assumption.
We write We have the following inequalities: We introduce now a family of ODEs which will be useful in investigating positive solutions of (ARD).
Proof.By the standard comparison results for ODEs lim sup where z i = a i − λ i (α i ) min x∈ Ω fi (x) ≥ 0. Lemma 2.4 and (2.9) imply that there exists ) + 1.From this it follows that the solutions of system (ARD) is defined for t ∈ [0, ∞).This proves (i).The proof of (ii) is now straightforward.Now we present the Vance and Coddinton result [28] which we use in the proof of the main theorem of this paper.
First we define c : [t 0 , ∞) → R, where t 0 > 0 to be a bounded continuous function where c, c > 0 are such that −c ≤ c(t) ≤ c for all t ≥ t 0 .. Assume moreover that there are L > 0 and β > 0 such that 1 L t+L t c(τ)dτ ≥ β for all t ≥ t 0 .
Proposition 2.6.For any positive solution ζ(t) of the initial value problem where the function c is as above and d is a positive constant there holds Assumptions (A3) and (A5) imply that there exist L > 0 and β > 0 such that then Proposition 2.6 implies that there exists δi > 0 which does not depend of the solution ξ i (t) such that δi ≤ lim inf

Average conditions for permanence
In this section we formulate the main theorem of this paper.We establish conditions which guarantee that the system (ARD) is permanent.Through this section we assume that ϕ i is normalized so that max x∈ Ω ϕ i (x) = 1 for i = 1, . . ., N.
In the case of the no-flux boundary conditions we have for t ≥ t 0 and x ∈ ∂Ω.Moreover Fix 1 ≤ i ≤ N. Now it suffices to apply Proposition 2.6 to equation (3.2) where and d = b ii ( 0 ).Now we show that the quantities appearing in Proposition 2.6 can be chosen independently of the solution u(t, x) at least for sufficiently large t.
It is easy to see that c is bounded from above with the bound independent of u(t, x).Take β and β such that Integrating inequality (2.7) from t to t + L we have that Hence for 1 ≤ j ≤ N and t ≥ t 1 and L ≥ L 0 .Inequality (2.10) implies that there exists L 0 > 0 such that for any positive solution of u(t, x) we can find t 0 ≥ 0 such that Therefore Then we have that for sufficiently large t.Now it suffices to apply Proposition 2.6 and the proof is completed.

Lower estimation of δ i
In this section we give a lower estimation on the numbers δ i which appear in the definition of permanence in terms of the parameters of system (ARD).The assumptions in this section are slightly stronger than (2.9).Through this section we assume that ϕ i is normalized so that max x∈ Ω ϕ i (x) = 1 for i = 1, . . ., N.
Theorem 4.1.Assume (A1) through (A5) and (AC).Assume, moreover, that for some then where β is positive constant satisfying and L > 0 is such that Proof.Fix a positive solution u(t, x) = (u 1 (t, x), . . ., u N (t, x)) of (ARD).Lemma 2.5 implies that for each there is t 0 ≥ 0 such that For each > 0 we define the positive solution ηi , t ≥ t 0 of the Similarly as in the main theorem we prove that u i (t, x) ≥ ηi (t)ϕ i (x) for all t ≥ t 0 , x ∈ Ω. Assume (4.2).Let 0 > 0 be so small that a i > λ i (α i ) max
∂ ∂n is the normal derivative.It follows from standard maximum principles for parabolic PDEs that there are functions γi : Dirichlet boundary conditions or Neumann boundary conditions.D i v i = ∂v i ∂n where n is the outward pointing normal vector and