Electronic Journal of Qualitative Theory of Differential Equations2018, No. 19, 1–18; https://doi.org/10.14232/ejqtde.2018.1.19 www.math.u-szeged.hu/ejqtde/

Permanence in N species nonautonomous competitivereaction–diffusion–advection system of Kolmogorov

type in heterogeneous environment.

Joanna BalbusB

Faculty of Pure and Applied Mathematics, Wrocław University of Technology,Wybrzeze Wyspianskiego 27, Wrocław, Poland

Received 6 June 2017, appeared 24 April 2018

Communicated by Michal Feckan

Abstract. One of the important concept in population dynamics is finding conditionsunder which the population can coexist. Mathematically formulation of this prob-lem we call permanence or uniform persistence. In this paper we consider N speciesnonautonomous competitive reaction–diffusion–advection system of Kolmogorov typein heterogeneous environment. Applying Ahmad and Lazer’s definitions of lower andupper averages of a function and using the sub- and supersolution methods for PDEswe give sufficient conditions for permanence in such models. We give also a lowerestimation on the numbers δi which appear in the definition of permanence in form ofparameters of system

∂ui∂t = ∇[µi∇ui − αiui∇ fi(x)] + fi(t, x, u1, . . . , uN)ui, t > 0, x ∈ Ω, i = 1, . . . , N,Diui = 0, t > 0, x ∈ ∂Ω, i = 1, . . . , N.

Keywords: advection, subsolution, lower average, permanence.

2010 Mathematics Subject Classification: 35K51, 92D25, 35B30, 35Q91, 35K61.

1 Introduction

A main problem in population dynamics is the long-term development of population. Uni-form persistence (sometimes also called permanence), coexistence and extinction describeimportant special types of asymptotic behavior of the solutions of associated model equa-tions. In this paper we consider the N species nonautonomous competitive reaction–diffusion–advection system of Kolmogorov type

∂ui

∂t= ∇[µi∇ui − αiui∇ fi(x)] + fi(t, x, u1, . . . , uN)ui, t > 0, x ∈ Ω, i = 1, . . . , N, (1.1)

BCorresponding author. Email: joanna.balbus@pwr.edu.pl. The author was supported as research project(S50082/K3301040110258/15).

2 J. Balbus

which is endowed in appropriate boundary conditions.In the context of ecology ui(t, x) denote the densities of the i-th species at time t and a

spatial location x ∈ Ω, Ω ⊂ Rn is a bounded habitat and

fi(x) = lim inft−s→∞

1t− s

∫ t

sfi(τ, x, 0, . . . , 0)dτ, (i = 1, . . . , N)

accounts for the local growth rate. If the environment is spatially heterogeneous i.e., fi(x) isnot 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 αi for 1 ≤ i ≤ N measuresthe rate at which the population moves up the gradient of fi(x). Through this paper we onlyconsider the case αi ≥ 0, for 1 ≤ i ≤ N i.e., the populations move up in the direction alongwhich 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 ofautonomous ODE sufficient conditions for permanence are given in a form of inequalitiesinvolving an interaction coefficients of the system (see e.g. [1]).

In [2] S. Ahmad and A. C. Lazer considered an N species nonautonomous competitiveLotka–Volterra system. The authors introduced a notion of upper and lower averages of afunction. They found sufficient conditions which guarantee that such system is permanentand globally attractive.

In [25] we extended their results on N species nonautonomous competitive system ofKolmogorov type.

The models of ODEs do not take into account spatial heterogeneity. They give the tem-poral changes in terms of the global population while partial differential equations give thetemporal changes at each point in space in terms of the local densities and the spatial gradi-ents. Dispersal of individuals has important effects from an ecological point of view and inthe 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 reaction–diffusion system of PDE

∂ui

∂t= µi∆ui + fi(t, x, u)ui, i = 1, . . . , N. (1.2)

The system (1.2) is an example of model of the population growth with unconditional dis-persal. Unconditional dispersal does not depend on habitat quality. This type of dispersalis investigated by many authors, see for example [3, 10, 14, 15, 17, 22, 23]. In [23] the authorsinvestigated uniform persistence for nonautonomous and randomly parabolic Kolmogorovsystems via the skew-product semiflows approach. They obtained sufficient conditions foruniform persistence in such systems in terms of Lyapunov exponents.

In [3] we studied N species nonautonomous reaction–diffusion Kolmogorov system withdifferent boundary conditions, either Dirichlet or Neumann or Robin boundary conditions.We gave sufficient conditions for permanence in such system. Those conditions are given in aform of inequalities involving time averages of intrinsic growth rates, interaction coefficients,migration rates and principal eigenvalues. In nature species do not move completely ran-domly. Their movements are a combination of both random and biased ones. Such modelsare called models with conditional dispersal. The most popular model which takes into ac-count some amounts of random motion and a purely directed movement dispersal strategy

Permanence in reaction–diffusion–advection system. 3

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 follow-ing form

∂u∂t = ∇ [∇u− αu∇m] + λu[m(x)− u] in Ω× (0, ∞),∂u∂n − αu ∂m

∂n = 0 on ∂Ω× (0, ∞).(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 boundaryacts as a reflecting barrier to the population i.e., no-flux across the boundary. Belgacem andCosner [4] studied (1.3) with both no-flux and Dirichlet boundary conditions. The authorsshowed 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 gradientof m(x) may be either beneficial or harmful to the population. The authors studied the effectof drift on the principal eigenvalues of certain elliptic operators. The eigenvalues determinewhether 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 shapeof the habitat of the population. In the case of convex habitat the movement in the directionof the gradient of the growth rate is always beneficial to the population. In the case of non-convex habitat such advection could be harmful to the population.

In [7] Chen et al. investigated a two species model of reaction–diffusion–advection∂u∂t = ∇[µ∇u− αu∇m(x)] + (m(x)− u− v)u,∂v∂t = ∇[ν∇v− βv∇m(x)] + (m(x)− u− v)v,

(1.4)

in Ω× (0, ∞) with no-flux boundary conditions

µ∂nu− αu∂nm = ν∂nv− βv∂nm = 0.

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 sameresources. They assumed also that m(x) is a nonconstant function. The resource is usuallyspatially 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 directedmovement upward along the gradient of m(x) (α(∇m)u and β(∇m)v). The authors showedthat if only one species has a strong tendency to move upward the environmental gradients thetwo species can coexist since one species mainly pursues resources at places of locally mostfavorable 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 overcrowdingof the whole population at places of locally most favorable environments which causes theextinction of the species with stronger biased movements.

In this paper we find sufficient conditions for uniform persistence in the N species nonau-tonomous competitive system of reaction–diffusion–advection. In contrast to [7, 13, 20, 21] weassume that all species have a different intrinsic per capita growth rates, and we take intoaccount the influence of the jth species of the growth rate of the ith species. The investigationof nonautonomous systems is of great importance biologically since in nature, many systems

4 J. Balbus

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 eigen-value of the eigenproblem (2.1). We also formulate auxiliary results on the behavior of thepositive solutions.

In Section 3 we state and prove the main theorem of this paper. We formulate averageconditions which guarantee that system (ARD) is permanent.

In Section 4 we formulate the stronger inequalities which give a lower bound on the pop-ulation densities in term of interaction coefficients of system (ARD).

2 Preliminaries

Consider a nonautonomous competitive N species model of reaction–diffusion–advection∂ui∂t = ∇[µi∇ui − αiui∇ fi(x)] + fi(t, x, u1, . . . , uN)ui, t > 0, x ∈ Ω, i = 1, . . . , N,

Diui = 0, t > 0, x ∈ ∂Ω, i = 1, . . . , N,(ARD)

where fi(x) = lim inft−s→∞1

t−s

∫ ts fi(τ, x, 0, . . . , 0)dτ are nonconstant functions for i = 1, . . . , N.

Ω ⊂ Rn is a bounded domain with the sufficiently smooth boundary ∂Ω, µi > 0 is a diffusionrate of the i-th species, αi ≥ 0 measure the rate at which the population moves up the gradientof the growth rate fi(x) of the i-th species and fi(t, x, u1, . . . , uN) is the local per capita growthrate of the i-th species.

We define the operator

L(ψi) =∂ψi

∂t+∇[µi∇ψi − αiψi∇ fi(x)] + fi(t, x, u)ψi.

Further we define the boundary operator Di which is either the Dirichlet operator

Di(ui) = ui on ∂Ω,

or the operator

Di(ui) = µi∂ui

∂n− αiui

∂ fi

∂non ∂Ω,

Denote by λi(αi) the principal eigenvalue of the eigenproblemµi∇2ϕi(x) + αi∇ fi(x)∇ϕi(x) = −λi(αi) fi(x)ϕi(x) on Ω,

Di ϕi = 0 on ∂Ω.(2.1)

In the case of Dirichlet boundary conditions it is known that (2.1) will always have a uniquepositive principal eigenvalue λ1

i (αi) 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 positiveprincipal eigenvalue αi(αi) characterized by having a positive eigenfunction if and only if∫

Ωfi(x)e

αiµi

fi(x) dx < 0.

Permanence in reaction–diffusion–advection system. 5

Definition 2.2. System (ARD) is permanent if there are positive constants δi, δi such that foreach positive solution u(t, x) = (u1(t, x), . . . , uN(t, x)) of (ARD)

δi ≤ lim inft→∞

ui(t, x)ϕi(x)

≤ lim supt→∞

ui(t, x)ϕi(x)

≤ δi, 1 ≤ i ≤ N,

where the limit is uniform in x ∈ Ω.

We introduce now a first assumption for a functions fi which guarantee the existence andthe uniqueness of local classical solutions to an initial value problem for (ARD).

(A1) fi : [0, ∞) × Ω × [0, ∞)N → R ( 1 ≤ i ≤ N), as well as their first derivatives ∂ fi∂t

(1 ≤ i ≤ N), ∂ fi∂uj

(1 ≤ i, j ≤ N) and ∂ fi∂xk

(1 ≤ i, k ≤ N) are continuous. Moreover,

the derivatives ∂ fi∂uj

(1 ≤ i, j ≤ N) are bounded and uniformly continuous on sets of the

form [0, ∞)× Ω× B where B is a bounded subset of [0, ∞)N .

(A1) is a standard assumption guaranteeing that for any sufficiently regular initial func-tion u0(x) = (u01(x), . . . , u0N(x)), x ∈ Ω there exists a unique maximally defined solutionu(t, x) = (u1(t, x), . . . , uN(t, x)) of (ARD), (t, x) ∈ [0, τmax)× Ω where τmax > 0, satisfying theinitial condition u(0, x) = u0(x). The solution u(t, x) is classical: the derivatives occurring inthe equations (resp. in the boundary conditions) are defined, and the equations (resp. theboundary conditions) are satisfied pointwise on (0, τmax)×Ω (resp. on (0, τmax)× ∂Ω). More-over, the derivatives ∂ui

∂t (i = 1, . . . , N) and ∂2ui∂xkxl

(1 ≤ k, l ≤ N, i = 1, . . . , N) are continuous on(0, τmax)× Ω.

We deal with the positive solutions of (ARD). By positive solution of (ARD) we meana solution u(t, x) = (u1(t, x), . . . , uN(t, x)) of (ARD) such that ui(t, x) > 0 for t ∈ (0, τmax),x ∈ Ω, i = 1, . . . , N. In other words, positive solutions correspond to initial functions u0(x) =(u01(x), . . . , u0N(x)) with u0i(x) ≥ 0, u0i 6= 0 for all i = 1, . . . , N.

For each 1 ≤ i ≤ N there holds

0 < infx∈Ω

ui(0, x)ϕi(x)

≤ supx∈Ω

ui(0, x)ϕi(x)

< ∞. (2.2)

Lemma 2.3. For any positive solution u(t, x) = (u1(t, x), . . . , uN(t, x)) of (ARD) there exist func-tions γ

i: (0, τmax)→ (0, ∞) and γi : (0, τmax)→ (0, ∞) such that

γi(t)ϕi(x) ≤ ui(t, x) ≤ γi(t)ϕi(x) (2.3)

for all t ∈ (0, τmax), x ∈ Ω, 1 ≤ i ≤ N.

Proof. Fix a positive solution u(t, x) = (u1(t, x), . . . , uN(t, x)) of (ARD). Denote by vi(t, x),1 ≤ i ≤ N, t ≥ 0, x ∈ Ω, the solution of the following boundary value problem

∂vi∂t = µi∇2vi + αi∇ fi∇vi, t > 0, x ∈ Ω,

Divi = 0 t > 0, x ∈ ∂Ω,

satisfying the initial condition vi(0, x) = ui(0, x), x ∈ Ω. Di denote Dirichlet boundary condi-tions or Neumann boundary conditions. Divi =

∂vi∂n where n is the outward pointing normal

6 J. Balbus

vector and ∂∂n is the normal derivative. It follows from standard maximum principles for

parabolic PDEs that there are functions γi : (0, ∞)→ (0, ∞) and γi

: (0, ∞)→ (0, ∞) such that

γi(t)ϕi(x) ≤ vi(t, x) ≤ γi(t)ϕi(x) (2.4)

for all t > 0, x ∈ Ω.For T ∈ (0, τmax) and 1 ≤ i ≤ N put

Mi = sup| fi(τ, x, u1(τ, x, u1(τ, x), . . . , uN(τ, x))| : τ ∈ [0, T], x ∈ Ω).

We prove that

eαiµi

f (x)−Mtvi(t, x) ≤ ui(t, x)

for t ∈ [0, T] and x ∈ Ω.We have

L(

vieαiµi

fi(x)−Mit)=

∂

∂t

(vie

αiµi

fi(x)−Mit)−∇

[µi∇

[vie

αiµi

fi(x)−Mit]− αivie

αiµi

fi(x)−Mit∇[ fi(x)]]

− fi(t, x, u)vieαiµi

fi(x)−Mit

=∂

∂t

(vie

αiµi

fi(x)−Mit)−Mivie

αiµi

fi(x)−Mit

− eαiµi

fi(x)−Mitvi(µi∇2vi + αi∇ fi(x)∇vi)− fi(t, x, u)vieαiµi

fi(x)−Mit

= vieαiµi

fi(x)−Mit(µi∇2vi + αi∇ fi(x)∇vi)−Mivieαiµi

fi(x)−Mit

− vieαiµi

fi(x)−Mit(µi∇2vi + αi∇ fi(x)∇vi)− fi(t, x, u1, . . . , uN)vieαiµi

fi(x)−Mit

= − vieαiµi

fi(x)−Mit(Mi + fi(t, x, u1, . . . , uN))

≤ 0

for t ∈ (0, T] and x ∈ Ω.In the case of Dirichlet boundary conditions we have

ui(t, x) ≥ vi(t, x)eαiµi

fi(x)−Mit (2.5)

for t > 0, x ∈ Ω and i = 1, . . . , N. In the case of no-flux boundary conditions we have

D(

vieαiµi

fi(x)−Mt)= µi

∂vi

∂n

(vie

αiµi

fi(x)−Mt)− αi

(vie

αiµi

fi(x)−Mt)∂ fi

∂n

= µieαiµi

fi(x)−Mt ∂vi

∂n= 0.

Again we have (2.5). In a similar way we show that

ui(t, x) ≤ vieαiµi

fi(x)+Mit (2.6)

for t ∈ (0, T], x ∈ Ω, i = 1, . . . , N.By (2.4), (2.5) and (2.6) we have the desired inequality.

Permanence in reaction–diffusion–advection system. 7

For i = 1, . . . , N, the function fi(t, x, 0, . . . , 0) is called the intrinsic growth rate of the i-th species. In [7] the authors assume that the two species have the same per capita growthrate. 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 quantityof resources available at the location x, where the favorable region x ∈ Ω : fi(x) > 0 actsas a resource and the unfavorable part x ∈ Ω : fi(x) < 0 is a sink region.

The assumption below is a standard boundedness assumption.

(A2) The functions [[0, ∞)× Ω 3 (t, x) 7→ fi(t, x, 0, . . . , 0) ∈ R], 1 ≤ i ≤ N are bounded.

We write

ai = inf fi(t, x, 0, . . . , 0) : t ≥ 0, x ∈ Ω,ai = sup fi(t, x, 0, . . . , 0) : t ≥ 0, x ∈ Ω.

For a bounded continuous function c : [0, ∞)→ R we define its lower average by

m[c] := lim inft−s→∞

1t− s

∫ t

sc(τ)dτ,

and its upper average by

M[c] := lim supt−s→∞

1t− s

∫ t

sc(τ)dτ.

Further we write

m[ fi] := lim inft−s→∞

1t− s

∫ t

sminx∈Ω

fi(τ, x, 0 . . . , 0)dτ,

M[ fi] := lim supt−s→∞

1t− s

∫ t

sminx∈Ω

fi(τ, x, 0 . . . , 0)dτ.

We have the following inequalities:

ai ≤ m[ fi] ≤ M[ fi] ≤ ai.

(A3) m[ fi] > 0, 1 ≤ i ≤ N,

(A4) ∂ fi∂uj

(t, x, u) ≤ 0 for all t ≥ 0, x ∈ Ω, u ∈ [0, ∞)N , 1 ≤ i, j ≤ N, i 6= j,

(A5) there exist bii > 0 such that ∂ fi∂ui

(t, x, u) ≤ −bii for all t ≥ 0, x ∈ Ω, u ∈ [0, ∞)N , 1 ≤ i ≤ N.

We introduce now a family of ODEs which will be useful in investigating positive solutionsof (ARD).

Let u(t, x) = (u1(t, x), . . . , uN(t, x)), t ∈ [0, τmax), be a positive solution of (ARD), where fisatisfies (A1), (A2), (A4) and (A5). For each 1 ≤ i ≤ N we define ξi(t), t ∈ [0, ∞) to be thepositive positive solution of the following initial value problemξ ′i(t) = (maxx∈Ω fi(t, x, 0, . . . , 0)− λi(αi)min

x∈Ωfi(x)− biiξi)ξi,

ξi(0) = supx∈Ω

ui(0,x)ϕi(x) e−

αiµi

fi(x)

.(2.7)

Note that, by (2.1), ξi(0) is finite for i = 1, . . . , N.

8 J. Balbus

Lemma 2.4. Assume that (A1), (A2), (A4) and (A5) hold. Then for any positive solution u(t, x) =

(u1(t, x), . . . , uN(t, x)) of (ARD), and any 1 ≤ i ≤ N, there holds

ui(t, x) ≤ ξi(t)eαiµi

fi(x)ϕi(x)

for t ∈ [0, τmax), x ∈ Ω where ξi(t) is the positive solution of (2.7).

Proof. Fix 1 ≤ i ≤ N. We prove that ξi(t)eαiµi

fi(x)ϕ(x) is a supersolution for ui(t, x).

By assumptions (A4) and (A5),

fi(t, x, u) ≤ maxx∈Ω

fi(t, x, 0, . . . , 0)− biiξi(t) t ∈ (0, τmax), x ∈ Ω, (2.8)

wherefi(t, x, u) := fi(t, x, u1(t, x), . . . , ui−1(t, x), ξi(t), ui+1(t, x), . . . , uN(t, x)).

Hence by (2.1), (2.7), (2.8)

L(ξi(t)eαiµi

fi(x)ϕi(x))

=∂

∂t(ξi(t)e

αiµi

fi(x)ϕi(x))−∇

[µi∇[ξi(t)e

αiµi

fi(x)ϕi(x)]− αiξi(t)e

αiµi

fi(x)ϕi(x)∇ fi(x)

]− fi(t, x, u)ξi(t)e

αiµi

fi(x)ϕi(x)

= ξi(t)eαiµi

fi(x)ϕi(x)

∂ξi

∂t− ξi(t)e

αiµi

fi(x) (αi∇ fi(x)∇ϕi(x) +∇2ϕi(x)

)− fi(t, x, u)ξi(t)e

αiµi

fi(x)ϕi(x)

=(2.1), (2.7)

ξi(t)eαiµi

fi(x)ϕi(x)

(−λi(αi)max

x∈Ωfi(x) + max

x∈Ωfi(t, x, 0, . . . , 0)− biiξi(t)

)+ ξi(t)e

αiµi

fi(x)λi(αi) fi(x)ϕi(x)− fi(t, x, u)ξi(t)e

αiµi

fi(x)ϕi(x)

≥(2.8)

ξi(t)eαiµi

fi(x)ϕi(x)

(−αi(λi(αi))min

x∈Ωf (x) + max

x∈Ωfi(t, x, 0, . . . , 0)− biiξi

+λi(αi) fi(x)− (maxx∈Ω

fi(t, x, 0, . . . , 0)− biiξi)

)≥ 0

for t ∈ (0, τmax) and x ∈ Ω.In the case of Dirichlet boundary conditions we have

D(ξi(t)eαiµi

fi(x)ϕi(x)) > 0, x ∈ ∂Ω, t ∈ (0, τmax).

In the case of no-flux boundary conditions we have

D(ξi(t)eαiµi

fi(x)ϕi(x)) = µi

∂

∂n(ξi(t)e

αiµi

fi(x)ϕi(x))− αiξi(t)e

αiµi

fi(x)ϕi(x)

∂ fi(x)∂n

= µiξi(t)eαiµi

fi(x) ϕi(x)∂n

= 0, x ∈ ∂Ω, t ∈ (0, τmax)

for t ∈ (0, τmax) and x ∈ ∂Ω. Moreover,

ξi(0)eαiµi

fi(x)ϕi(x) = sup

x∈Ω

ui(0, x)ϕi(x)

e−αiµi

fi(x)

eαiµi

fi(x)ϕi(x) ≥ ui(0, x)

Permanence in reaction–diffusion–advection system. 9

for x ∈ Ω. Thereforeui(t, x) ≤ ξi(t)e

αiµi

fi(x)ϕi(x)

for all t ∈ (0, τmax) and x ∈ Ω.

Lemma 2.5. Assume (A1)–(A5) and ai − λi(αi)minx∈Ω fi(x) ≥ 0. Then for any maximally definedpositive solution u(t, x) = (u1(t, x), . . . , uN(t, x)) of (ARD) we have

(i) τmax = ∞, and

(ii) lim supt→∞ ui(t, x) ≤ zibii

, where zi = ai − λi(αi)minx∈Ω fi(x) where the limit is uniformly inx ∈ Ω.

Proof. By the standard comparison results for ODEs

lim supt→∞

ξi(t) ≤zi

bii< ∞, (2.9)

where zi = ai − λi(αi)minx∈Ω fi(x) ≥ 0. Lemma 2.4 and (2.9) imply that there exists t1 ≥ 0such that C(Ω) norm of ui(t, x) is bounded on [t1, τmax) by ( zi

bii) + 1. From this it follows that

the solutions of system (ARD) is defined for t ∈ [0, ∞). This proves (i). The proof of (ii) is nowstraightforward.

Now we present the Vance and Coddinton result [28] which we use in the proof of themain theorem of this paper.

First we define c : [t0, ∞) → R, where t0 > 0 to be a bounded continuous function wherec, c > 0 are such that −c ≤ c(t) ≤ c for all t ≥ t0.. Assume moreover that there are L > 0 andβ > 0 such that

1L

∫ t+L

tc(τ)dτ ≥ β

for all t ≥ t0.

Proposition 2.6. For any positive solution ζ(t) of the initial value problemζ ′ = (c(t)− dζ)ζ,

ζ(t0) = ζ0 > 0,

where the function c is as above and d is a positive constant there holds

β

de−L(c+β) ≤ lim inf

t→∞ζ(t) ≤ lim sup

t→∞ζ(t) ≤ c

d.

Assumptions (A3) and (A5) imply that there exist L > 0 and β > 0 such that

1L

∫ t+L

tmaxx∈Ω

fi(τ, x, 0, . . . , 0)dτ ≥ β

for all t ≥ 0 and 1 ≤ i ≤ N.If we let c(t) = maxx∈Ω fi(t, x, 0, . . . , 0), di = bii and ai > λi(αi)minx∈Ω fi(x) then Propo-

sition 2.6 implies that there exists δi > 0 which does not depend of the solution ξi(t) suchthat

δi ≤ lim inft→∞

ξi(t) ≤ lim supt→∞

ξi(t) ≤ai − λi(αi)min

x∈Ωfi(x)

bii. (2.10)

10 J. Balbus

For 1 ≤ i, j ≤ N and ε ≥ 0 we define bij(ε) as the supremum− ∂ fi

∂uj(t, x, u) : t ≥ 0, x ∈ Ω, u ∈

0,a1 − λ1(α1)min

x∈Ωf1(x)

b11+ ε

× · · · ×

0,aN − λN(αN)min

x∈ΩfN(x)

bNN+ ε

.

Instead of bij(0) we write bij. Assumptions (A4) and (A5) imply that bij ≥ 0, 1 ≤ i, j,≤ N. By(A1) it follows that bij(ε) < ∞ and limε→0+ bij(ε) = bij for 1 ≤ i, j ≤ N.

3 Average conditions for permanence

In this section we formulate the main theorem of this paper. We establish conditions whichguarantee that the system (ARD) is permanent. Through this section we assume that ϕi isnormalized so that maxx∈Ω ϕi(x) = 1 for i = 1, . . . , N.

Theorem 3.1. Assume (A1) through (A5) and ai > λi(αi)minx∈Ω

fi(x) for i = 1, . . . , N. If

m[ fi] > λi(αi)maxx∈Ω

fi(x) +N

∑j=1j 6=i

eαjµj

maxx∈Ω

f j(x)bij(M[ f j]− λj(αj)min

x∈Ωf j(x))

bjj(3.1)

for all 1 ≤ i ≤ N then system (ARD) is permanent.

Proof. Let ε0 > 0 be such that

m[ fi] > λi(αi)maxx∈Ω

fi(x) +N

∑j=1j 6=i

eαjµj

maxx∈Ω

f j(x)bij(ε0)(M[ f j]− λj(αj)min

x∈Ωf j(x))

bjj

for all 1 ≤ i ≤ N.Fix a positive solution u(t, x) = (u1(t, x), . . . , uN(t, x)) of (ARD).Let ξi(t), t ≥ 0, 1 ≤ i ≤ N be the solution of (2.5) corresponding to u(t, x). Let t0 ≥ 0 be

such that ξi(t) ≤ ai−λi(αi)minx∈Ω fi(x)bii

+ ε2 for all t ≥ t0, 1 ≤ i ≤ N.

Denote by ηi(t), 1 ≤ i ≤ N, t ≥ t0 the positive solution of the initial value problem

η′i(t) =

minx∈Ω

fi(t, x, 0, . . . , 0)− λi(αi)maxx∈Ω

fi(x)− bii(ε0)ηi(t)

−N

∑j=1j 6=i

bij(ε0)ξ j(t)eαjµj

maxx∈Ω

f j(x)

ηi,

ηi(t0) = infx∈Ω

ui(t0, x)

ϕi(x)e−

αiµi

fi(x)

.

(3.2)

We prove that

ui(t, x) ≥ ηi(t)eαiµi

fi(x)ϕi(x)

Permanence in reaction–diffusion–advection system. 11

for all t ≥ t0 and x ∈ Ω.By Lemma 2.4 it follows that

ui(t, x) ≤ ξi(t)eαiµi

fi(x)ϕi(x) ≤ ξi(t)e

αiµi

maxx∈Ω

fi(x)

for t ≥ t0 and x ∈ Ω.Assumption (A1) and Lemma 2.1 imply that

fi(t, x, u) ≥ minx∈Ω

fi(, x, 0, . . . , 0)− bii(ε0)ηi(t)ϕi(x)−N

∑j=1j 6=i

uj(t, x)

≥ minx∈Ω

fi(, x, 0, . . . , 0)− bii(ε0)ηi(t)ϕi(x)−N

∑j=1j 6=i

ξ j(t)eαiµi

maxx∈Ω

fi(x),

(3.3)

where

fi(t, x, u) := fi(t, x, ui(t, x), . . . , ui−1(t, x), ηi(t)ϕi(x), ui+1(t, x), . . . , uN(t, x)).

By (2.1), (3.2), (3.3) we have

L(ηi(t)eαiµi

fi(x)ϕi(x))

=∂

∂t(ηi(t)e

αiµi

fi(x)ϕi(x))−∇

[µi∇[ηi(t)e

αiµi

fi(x)ϕi(x)]− αiηi(t)e

αiµi

fi(x)ϕi(x)∇ fi(x)

]− fi(t, x, u)ηi(t)e

αiµi

fi(x)ϕi(x)

= ηi(t)eαiµi

fi(x)ϕi(x)

∂ηi

∂t− ηi(t)e

αiµi

fi(x) (αi∇ fi(x)∇ϕi(x) +∇2ϕi(x)

)− fi(t, x, u)ηi(t)e

αiµi

fi(x)ϕi(x)

=(2.1), (3.2)

ηi(t)eαiµi

fi(x)ϕi(x)

− λi(αi)minx∈Ω

fi(x) + minx∈Ω

fi(t, x, 0, . . . , 0)− bii(ε)ηi(t)

−N

∑j=1j 6=i

bij(ε)ξ j(t)eαjµj

maxx∈Ω

f j(x)

+ ηi(t)eαiµi

fi(x)λi(αi) fi(x)ϕi(x)

− fi(t, x, u)ξi(t)eαiµi

fi(x)ϕi(x)

≤(3.3)

ηi(t)eαiµi

fi(x)ϕi(x)

− λi(αi)maxx∈Ω

fi(x) + minx max∈Ω

fi(t, x, 0, . . . , 0)− biiηi

−N

∑j=1j 6=i

bij(ε)ξ j(t)eαjµj

maxx∈Ω

f j(x)+ λi(αi) fi(x)

−

minx∈Ω

fi(t, x, 0, . . . , 0)− bii(ε)ηi −N

∑j=1j 6=i

bij(ε)ξ j(t)eαjµj

maxx∈Ω

f j(x)

≤ 0

12 J. Balbus

for t ≥ t0 and x ∈ Ω. In the case of the Dirichlet boundary conditions we have

D(ηi(t)eαiµi

fi ϕi(x)) > 0, x ∈ ∂Ω, t ∈ (0, τmax).

In the case of the no-flux boundary conditions we have

D(ηi(t)eαiµi

fi(x)ϕi(x)) = µi

∂

∂n(ηi(t)e

αiµi

fi(x)ϕi(x))− αiηi(t)e

αiµi

fi(x)ϕi(x)

∂ fi(x)∂n

= µiηi(t)eαiµi

fi(x) ∂ϕi(x)∂n

= 0

for t ≥ t0 and x ∈ ∂Ω.Moreover

ηi(t0)eαiµi

fi(x)ϕi(x) = inf

x∈Ω

ui(t0, x)

ϕi(x)e−

αiµi

fi(x)

eαiµi

fi(x)ϕi(x) ≤ ui(t0, x)

for x ∈ Ω.

Fix 1 ≤ i ≤ N. Now it suffices to apply Proposition 2.6 to equation (3.2) where

c(t) = minx∈Ω

fi(t, x, 0, . . . , 0)− λi(αi)maxx∈Ω

fi(x)−N

∑j=1j 6=i

bij(ε0)ξ j(t)eαjµj

maxx∈Ω

f j(x)

and d = bii(ε0).Now we show that the quantities appearing in Proposition 2.6 can be chosen indepen-

dently 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

0 < β < β′ < m[ fi]− λi(αi)maxx∈Ω

fi(x)−N

∑j=1j 6=i

eαjµj

maxx∈Ω

f j(x)bij(ε0)(M[ f j]− λj(αj)min

x∈Ωf j(x))

bjj. (3.4)

Integrating inequality (2.7) from t to t + L we have that

bjj

L

∫ t+L

tξ j(t)dt =

1L

∫ t+L

tmaxx∈Ω

f j(t, x, 0, . . . , 0)dt− λj(αj)minx∈Ω

f j(x)−| ln ξ j(t + L)− ln ξ j(t)|

L.

Hence

bjj M[ξ j] = M[ f j]− λj(αj)minx∈Ω

f j(x)−| ln ξ j(t + L)− ln ξ j(t)|

Lfor 1 ≤ j ≤ N and t ≥ t1 and L ≥ L0.

Inequality (2.10) implies that there exists L0 > 0 such that for any positive solution ofu(t, x) we can find t0 ≥ 0 such that

| ln ξ j(t + L)− ln ξ j(t)|L

<(β′ − β)bjj

N maxk 6=j bkl(ε0). (3.5)

Therefore

M[ξ j] =M[ f j]

bjj−

λj(αj)minx∈Ω

f j(x)

bjj− 1

bjj

| ln ξ j(t + L)− ln ξ j(t)|L

.

Permanence in reaction–diffusion–advection system. 13

Then we have that

m

minx∈Ω

fi(t, x, 0, . . . , 0)− λi(αi)maxx∈Ω

fi(x)−N

∑j=1j 6=i

bij(ε0)ξ j(t))eαjµj

maxx∈Ω

f j(x)

≥ m

[minx∈Ω

fi(t, x, 0, . . . , 0)]− λi(αi)max

x∈Ωfi(x)−

N

∑j=1j 6=i

bij(ε0)m[ξ j(t)]eαjµj

maxx∈Ω

f j(x)

= m[

minx∈Ω

fi(t, x, 0, . . . , 0)]− λi(αi)max

x∈Ωfi(x)

−N

∑j=1j 6=i

bij(ε0))eαjµj

maxx∈Ω

f j(x)

M[ fi]

bii−

λi(αi)minx∈Ω

fi(x)

bii− 1

bii

| ln ξ j(t + L)− ln ξ j(t)|L

= m[ fi]− λi(αi)max

x∈Ωfi(x)−

N

∑j=1j 6=i

eαjµj

maxx∈Ω

f j(x)(

bij(ε0)

bjj(M[ fi]− λi(αi)min

x∈Ωfi(x)

)

+N

∑j=1j 6=i

eαjµj

maxx∈Ω

f j(x) bij(ε0)

bjj

| ln ξ j(t + L)− ln ξ j(t)|L

> β

for sufficiently large t. Now it suffices to apply Proposition 2.6 and the proof is completed.

4 Lower estimation of δi

In this section we give a lower estimation on the numbers δi which appear in the definitionof permanence in terms of the parameters of system (ARD). The assumptions in this sectionare slightly stronger than (2.9). Through this section we assume that ϕi is normalized so thatmaxx∈Ω ϕi(x) = 1 for i = 1, . . . , N.

Theorem 4.1. Assume (A1) through (A5) and (AC). Assume, moreover, that for some 1 ≤ i ≤ N astronger inequality

m[ fi] > λi(αi)maxx∈Ω

fi(x) +N

∑j=1j 6=i

eαjµj

maxx∈Ω

f j(x)bij(aj − λj(αj)min

x∈Ωf j(x))

bjj(4.1)

holds.

(i) If

ai > λi(αi)maxx∈Ω

fi(x) +N

∑j=1j 6=i

eαjµj

maxx∈Ω

f j(x)bij(aj − λj(αj)min

x∈Ωf j(x))

bjj, (4.2)

then

δi ≥1bii

ai − λi(αi)maxx∈Ω

fi(x)−N

∑j=1j 6=i

eαjµj

maxx∈Ω

f j(x)bij(aj − λj(αj)min

x∈Ωf j(x))

bjj

.

14 J. Balbus

(ii) If

ai ≤ λi(αi)maxx∈Ω

fi(x) +N

∑j=1j 6=i

eαjµj

maxx∈Ω

f j(x)bij(aj − λj(αj)min

x∈Ωf j(x))

bjj, (4.3)

thenδi ≥

β

biiexp(−L(m[ fi]− ai)),

where β is positive constant satisfying

β < m[ fi]− λi(αi)maxx∈Ω

fi(x)−N

∑j=1j 6=i

eαjµj

maxx∈Ω

f j(x)bij(aj − λj(αj)min

x∈Ωf j(x))

bjj(4.4)

and L > 0 is such that

1L

∫ t+L

tminx∈Ω

fi(τ, x, 0, . . . , 0)dτ > β + λi(αi) f(x) +N

∑j=1j 6=i

eαjµj

maxx∈Ω

f j(x)bij(aj − λj(αj)min

x∈Ωf j(x))

bjj

for all t ≥ 0.

Proof. Fix a positive solution u(t, x) = (u1(t, x), . . . , uN(t, x)) of (ARD). Lemma 2.5 impliesthat for each ε there is t0 ≥ 0 such that

ui(t, x) ≤ai − λi(αi)min

x∈Ωfi(x)

bii+

ε

2

for t ≥ t0 x ∈ Ω, 1 ≤ i ≤ N. For each ε > 0 we define the positive solution ηi, t ≥ t0 of theIVP

η′i(t) =

minx∈Ω

fi(t, x, 0, . . . , 0)− λi(αi)maxx∈Ω

fi(x)− bii(ε)ηi(t)

−N

∑j=1j 6=i

eαjµj

maxx∈Ω

f j(x)bij(ε)

aj − λj(αj)minx∈Ω

f j

bjj+ ε

ηi

ηi(t0) = infx∈Ω

ui(t0, x)

ϕi(x)e−

αiµi

fi(x)

.

Similarly as in the main theorem we prove that ui(t, x) ≥ ηi(t)ϕi(x) for all t ≥ t0, x ∈ Ω.Assume (4.2).

Let ε0 > 0 be so small that

ai > λi(αi)maxx∈Ω

fi(x) +N

∑j=1j 6=i

eαjµj

maxx∈Ω

f j(x)bij(ε0)

aj − λj(αj)minx∈Ω

f j(x)

bjj+ ε0

.

For each ε ∈ (0, ε0], by standard comparison principle for ODEs there holds

lim inft→∞

ηi(t) ≥1

bii(ε)

ai − λi(αi)maxx∈Ω

fi(x)−N

∑j=1j 6=i

eαjµj

maxx∈Ω

f j(x)bij(ε)

aj − λj(αj)minx∈Ω

f j(x)

bjj+ ε

.

Permanence in reaction–diffusion–advection system. 15

If ε→ 0, then

lim inft→∞

ηi(t) ≥1bii

ai − λi(αi)maxx∈Ω

fi(x)−N

∑j=1j 6=i

eαjµj

maxx∈Ω

f j(x)bij

aj − λj(αj)minx∈Ω

f j(x)

bjj

.

Now we assume (4.3). Let β > 0 be such that

β < m[ fi]− λi(αi)maxx∈Ω

fi(x)−N

∑j=1j 6=i

eαjµj

maxx∈Ω

f j(x)bij(ε0)(aj − λj(αj)min

x∈Ωf j(x))

bjj

and take L > 0 such that

1L

∫ t+L

tminx∈Ω

fi(t, x, 0, . . . , 0) > β + λi(αi)maxx∈Ω

fi(x)−N

∑j=1j 6=i

bij(ε0)(aj − λj(αj)minx∈Ω

f j(x))

bjj

for all t ≥ 0. Let ε0 > 0 be so small that

β < m[ fi]− λi(αi)maxx∈Ω

fi(x)−N

∑j=1j 6=i

eαjµj

maxx∈Ω

f j(x)bij(ε0)

aj − λj(αj)minx∈Ω

f j(x)

bjj+ ε

.

For ε ∈ (0, ε0] put

c(t) := minx∈Ω

fi(t, x, 0, . . . , 0)− λi(αi)maxx∈Ω

fi(x)

−N

∑j=1j 6=i

eαjµj

maxx∈Ω

f j(x)bij(ε0)

aj − λj(αj)minx∈Ω

f j(x)

bjj+ ε

.

It is easy to see thatc(t) ≥ −c

for all t ≥ t0, where

c = −ai + λi(αi)maxx∈Ω

fi(x) +N

∑j=1j 6=i

eαjµj

maxx∈Ω

f j(x)bij

aj − λj(αj)minx∈Ω

f j(x)

bjj+ ε

.

Now it suffices to apply Proposition 2.6 which gives the following inequality

lim inft→∞

ηi(t) ≥β

bii(ε0) exp

− L

β− ai + λi(αi)max fi(x)

+N

∑j=1j 6=i

eαjµj

maxx∈Ω

f j(x)bij(ε0)

(aj − λj(αj)min f j(x)

bjj+ ε

) .

16 J. Balbus

If ε→ 0 then

lim inft→∞

ui(t, x)ϕ(x)

≥ β

biiexp

−L

β− ai + λi(αi)max fi(x) +N

∑j=1j 6=i

eαjµj

maxx∈Ω

f j(x)bij

(aj − λj(αj)min f j(x)

bjj

) .

Therefore from (4.4)

lim inft→∞

ui(t, x)ϕ(x)

≥ β

biiexp(−L(m[ fi]− ai)).

Acknowledgment

The author is indebted to the anonymous referee for valuable comments and suggestions.

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