Competitive-cooperative models with various diffusion strategies
aa r X i v : . [ m a t h . D S ] J un Competitive-cooperative models with various diffusion strategies
E. Braverman , Md. Kamrujjaman a Department of Mathematics and Statistics, University of Calgary, 2500 University Drive N. W.,Calgary, AB T2N 1N4, Canada
Abstract
The paper is concerned with different types of dispersal chosen by competing species. We introducea model with the diffusion-type term ∇· [ a ∇ ( u/P )] which includes some previously studied systemsas special cases, where a positive space-dependent function P can be interpreted as a chosen disper-sal strategy. The well-known result that if the first species chooses P proportional to the carryingcapacity while the second does not then the first species will bring the second one to extinction, isalso valid for this type of dispersal. However, we focus on the case when the ideal free distributionis attained as a combination of the two strategies adopted by the two species. Then there is aglobally stable coexistence equilibrium, its uniqueness is justified. If both species choose the samedispersal strategy, non-proportional to the carrying capacity, then the influence of higher diffusionrates is negative, while of higher intrinsic growth rates is positive for survival in a competition.This extends the result of [J. Math. Biol. (1) (1998), 61–83] for the regular diffusion to a moregeneral type of dispersal. Keywords: ideal free distribution, ideal free pair, dispersal strategy, global attractivity, system ofpartial differential equations, coexistence
1. Introduction
The study of population dynamics started with the description of the total population size andits dynamics. However, taking into account the spatial structure is essential for understanding ofinvasion, survival and extinction of species. In order to involve movements in spatially distributedsystems, the dispersal strategy should be specified. The ideas of spreading over the domain toavoid overcrowding combined with advection towards higher available resources were incorporatedin the model in [1, 5, 6, 7, 8]. Dispersal design in [6] was based on the notion of the ideal freedistribution, i.e. such distribution that any movement in an ideally distributed system will decreasethe fitness of moving individuals. An ideal free distribution in a temporally constant but spatiallyheterogeneous environment is expected to be a solution of the system. There were several possibleapproaches to model dispersal in such a way that the ideal free distribution is a stationary solutionof the equation, see, for example, [3, 6, 7, 8].Most of the previous research [6, 7, 8, 9, 11, 12] was focused on evolutionarily stable strategieswhich provide advantages in a competition. This allowed to answer the question: what parametersor strategies should a spatially distributed population choose so that its habitat cannot be invaded
Email address: [email protected] (E. Braverman )
Preprint submitted to Computers & Mathematics with Applications August 10, 2018 y a species choosing alternative strategy and having other parameters? The effect of environ-ment heterogeneity was studied in [15]. The case when a preferred diffusion strategy alleviated anegative effect of less efficient resources exploitation leading to possible coexistence was recentlystudied in [2]. However, the question how spatial distribution can promote coexistence comparedto homogeneous environment got much less attention. Coexistence of various species relying onthe same resources is quite common in nature. Survival of a certain population can be secured if itfinds a certain niche in the resource consumption. Adaptation of different parts of the habitat (forexample, shallow waters and deep waters) by two (or more) competing for resources populationscan lead to coexistence.The purpose of this paper is two-fold.1. The first goal is to unify different approaches to diffusion strategies and dispersal, for example,the advection to better environments in [5] and the model where not the population densitybut its ratio to locally available resources diffuses [3]. A type of dispersal which includes [3, 5]is introduced in Section 2.2. The second goal is to consider the case when dispersal strategies guarantee coexistence. Ifthe carrying capacity of the environment is a combination of these strategies, the ideal freedistribution is attained at the coexistence stationary solution where the first population canbe more abundant in some areas of the habitat, while the second one can be dense in otherareas. Altogether, the two populations use the resources in an optimal way, and the sum oftwo densities coincides with the carrying capacity at any place of the spatial domain. Thiscoexistence equilibrium is unique and globally attractive, see Section 3.In addition, we extend the results of [9] to the case when the diffusion is not necessarily regularbut the diffusion strategy is the same for both species, and it is different from [3] where, as thediffusion coefficient tends to infinity, the solution tends to the carrying capacity which coincideswith the ideal free distribution. If this is the case, higher diffusion coefficient leads to a disadvantagein a competition. However, Section 4 also gives an alternative interpretation to this result: higherintrinsic growth rates (assuming the two have the same dispersal strategy) give an evolutionaryadvantage. Finally, Section 5 includes discussion of the results obtained and states some openquestions.
2. Dispersal Modeling
When choosing the type of diffusion, we focus on the ideal free distribution which is expectedto be a solution of the equation, in the absence of other species. A related idea is that there is amovement towards higher available resources, not just lower densities, as the regular diffusion ∆ u would suggest. For our modeling, we assume the null hypothesis that not u but u/P is subject toadvection or diffusion, where P is a diffusion strategy chosen by a species whose density at time t and point x is u ( t, x ). Here P is a positive and smooth in the domain Ω function. Assumingspace-dependent rate of advection a ( x ), we consider ∇ · (cid:20) a ( x ) ∇ (cid:18) u ( t, x ) P ( x ) (cid:19)(cid:21) = ∇ · (cid:20) a ( x ) P ( x ) (cid:18) ∇ u ( t, x ) − u ( t, x ) ∇ PP (cid:19)(cid:21) . (2.1)The space-distributed P is treated as a diffusion strategy in the following sense: if a ( x ) → + ∞ ,the density u ( t, x ) would be proportional to P , whatever growth law we choose. Accepting (2.1) as2 dispersal model, we consider the following system of two competing species obeying the logisticgrowth rule, with the Neumann boundary conditions ∂u∂t = ∇ · (cid:20) a ( x ) P ( x ) (cid:16) ∇ u ( t, x ) − u ( t, x ) ∇ PP (cid:17)(cid:21) + r ( x ) u ( t, x ) (cid:18) − u ( t, x ) + v ( t, x ) K ( x ) (cid:19) ,∂v∂t = ∇ · (cid:20) a ( x ) Q ( x ) (cid:16) ∇ v ( t, x ) − v ( t, x ) ∇ QQ (cid:17)(cid:21) + r ( x ) v ( t, x ) (cid:18) − u ( t, x ) + v ( t, x ) K ( x ) (cid:19) ,t > , x ∈ Ω ,∂u∂n − uP ∂P∂n = ∂v∂n − vQ ∂Q∂n = 0 , x ∈ ∂ Ω ,u (0 , x ) = u ( x ) , v (0 , x ) = v ( x ) , x ∈ Ω . (2.2)Let us note that, for smooth positive P and Q , the boundary conditions in (2.2) are equivalent to ∂∂n (cid:16) uP (cid:17) = 0 , ∂∂n (cid:18) vQ (cid:19) = 0 , x ∈ ∂ Ω . (2.3)The three most common particular cases are outlined below:1. If either both P and a or both Q and a are constant, then either the first or the secondequation incorporates a regular diffusion term d ∆ u or d ∆ v , where d = a /P or d = a /Q .2. If a is space-independent, in the first equation of (2.2) we obtain the type of dispersal∆( u/P ). In the particular case when P ≡ K (or P is proportional to K ) we have the term∆( u/K ), first introduced in [3] and later considered in [10]-[13]. If Q is constant, while a isproportional to 1 /K , we have the dispersal type ∇ · ( K ∇ v ) which was considered in [11, 12].3. If a = µ P , ln P = µ K , where µ i , i = 1 , r = K , we obtainthe directed advection model of the type ∂u∂t = ∇ · [ µ ∇ u − αu ∇ K ] + u ( K − u ) , (2.4)where µ = µ , α = µ µ . Equation (2.4) was considered in [1, 5, 6, 7, 8], see also referencestherein.Thus, (2.2) generalizes most of earlier considered dispersal strategies. However, it should bementioned that, for constant a /P , there are publications where P is not assumed to be positiveeverywhere on Ω.
3. Directed Diffusion Competition Model
We assume that the domain Ω is an open bounded region in R n with ∂ Ω ∈ C β , β >
0, thefunctions r ( x ), a i ( x ), i = 1 , P ( x ), Q ( x ) and K ( x ) are continuous and positive on Ω; moreover, a, P, Q ∈ C (Ω).Next, let us proceed to the study of stationary solutions (positive equilibria) of (2.2). Problem(2.2) is a monotone dynamical system [4, 18, 20]. If all equilibrium solutions but one are unstable,we would be able to conclude that the remaining equilibrium is globally asymptotically stable. Westart with the trivial equilibrium, and, similarly to [11, 12], verify the following result.3 emma . The zero solution of (2.2) is unstable; moreover, it is a repeller.Let functions u ∗ and v ∗ be solutions of the single-species stationary models corresponding tothe first and the second equations in (2.2) ∇ · (cid:20) a ( x ) ∇ (cid:18) u ∗ ( x ) P ( x ) (cid:19)(cid:21) + r ( x ) u ∗ ( x ) (cid:18) − u ∗ ( x ) K ( x ) (cid:19) = 0 , x ∈ Ω , ∂ ( u ∗ /P ) ∂n = 0 , x ∈ ∂ Ω , (3.1) ∇ · (cid:20) a ( x ) ∇ (cid:18) v ∗ ( x ) Q ( x ) (cid:19)(cid:21) + r ( x ) v ∗ ( x ) (cid:18) − v ∗ ( x ) K ( x ) (cid:19) = 0 , x ∈ Ω , ∂ ( v ∗ /Q ) ∂n = 0 , x ∈ ∂ Ω , (3.2)respectively.In future, we will need the following two auxiliary statements. Lemma . Let u ∗ be a positive solution of (3.1), then Z Ω r ( x ) P ( x ) (cid:18) u ∗ ( x ) K ( x ) − (cid:19) dx = Z Ω a ( x ) |∇ ( u ∗ /P ) | ( u ∗ /P ) dx. (3.3)If P ( x ) and K ( x ) are linearly independent on Ω, then Z Ω r ( x ) P ( x ) (cid:18) u ∗ ( x ) K ( x ) − (cid:19) dx > . (3.4) Proof.
Since u ∗ > P ( x ) > x ∈ Ω, dividing equation (3.1) by u ∗ /P , we obtain ∇ · [ a ∇ ( u ∗ /P )]( u ∗ /P ) + r ( x ) P ( x ) (cid:18) − u ∗ ( x ) K ( x ) (cid:19) = 0 , x ∈ Ω , ∂ ( u ∗ /P ) ∂n = 0 , x ∈ ∂ Ω (3.5)Integrating (3.5) over the domain Ω using boundary conditions in (3.5), we have Z Ω a |∇ ( u ∗ /P ) | ( u ∗ /P ) dx + Z Ω r ( x ) P ( x ) (cid:18) − u ∗ ( x ) K ( x ) (cid:19) dx = 0 (3.6)Therefore Z Ω r ( x ) P ( x ) (cid:18) u ∗ ( x ) K ( x ) − (cid:19) dx = Z Ω a ( x ) |∇ ( u ∗ /P ) | ( u ∗ /P ) dx > , (3.7)unless u ∗ /P is identically equal to a positive constant. However, substituting u ∗ ( x ) = cP ( x ) into(3.1), we obtain cP ( x ) ≡ K ( x ), x ∈ Ω, which contradicts to our assumption that P and K arelinearly independent on Ω.A similar result is valid for v ∗ whenever Q and K are linearly independent.The analogues of the following statements for less general diffusion types were obtained in[11, 12], for completeness we present the proof here. Lemma . Suppose that u ∗ is a positive solution of (3.1), while P ( x ) and K ( x ) satisfy ∇ · [ a ( x ) ∇ ( K ( x ) /P ( x ))] Z Ω r ( x ) K ( x ) dx > Z Ω r ( x ) u ∗ ( x ) dx. (3.8)4 roof. Substituting u = u ∗ and v ≡ Z Ω ru ∗ (cid:18) − u ∗ K (cid:19) dx = − Z Ω rK (cid:18) − u ∗ K (cid:19) dx + Z Ω rK (cid:18) − u ∗ K (cid:19) dx. Since the first integral in the right-hand side is non-positive, the second integral is non-negative.Moreover, it is positive unless u ∗ ≡ K which would imply ∇ · [ a ( x ) ∇ ( K ( x ) /P ( x ))] ≡ ∇ · [ a ( x ) ∇ ( K ( x ) /Q ( x ))] Z Ω r ( x ) K ( x ) (cid:18) − v ∗ ( x ) K ( x ) (cid:19) dx > . (3.9)Similarly to Lemma 3, the following result is justified. Lemma . Suppose that ( u s , v s ) is a positive stationary solution of (2.2), such that u s ( x ) + v s ( x ) K ( x ). Then Z Ω r ( x ) K ( x ) (cid:18) − u s ( x ) + v s ( x ) K ( x ) (cid:19) dx > . (3.10) Lemma . Suppose that P , K and Q , K are two pairs of linearly independent on Ω functions, while K ( x ) ≡ αP + βQ for some α > β >
0. Then the semi-trivial steady state ( u ∗ ( x ) ,
0) of (2.2) isunstable.
Proof.
Consider the eigenvalue problem associated with the second equation in (2.2) around theequilibrium ( u ∗ ( x ) , ∇ · (cid:20) a ( x ) ∇ (cid:18) ψ ( x ) Q ( x ) (cid:19)(cid:21) + r ( x ) ψ ( x ) (cid:18) − u ∗ ( x ) K ( x ) (cid:19) = σψ ( x ) , x ∈ Ω , ∂ ( ψ/Q ) ∂n = 0 , x ∈ ∂ Ω (3.11)The principal eigenvalue of (3.11) is defined as [4] σ = sup ψ =0 ,ψ ∈ W , − Z Ω a |∇ ( ψ/Q ) | dx + Z Ω r ( x ) ψ Q (cid:18) − u ∗ K (cid:19) dx , Z Ω ψ Q dx.
Choosing ψ ( x ) = √ βQ ( x ) and denoting M := Z Ω βQ ( x ) dx, we observe that the principaleigenvalue is not less than σ ≥ M Z Ω r ( x ) βQ ( x ) (cid:18) − u ∗ ( x ) K ( x ) (cid:19) dx = 1 M Z Ω r ( x )( K ( x ) − αP ( x )) (cid:18) − u ∗ ( x ) K ( x ) (cid:19) dx = 1 M Z Ω r ( x ) K ( x ) (cid:18) − u ∗ ( x ) K ( x ) (cid:19) dx + αM Z Ω r ( x ) P ( x ) (cid:18) u ∗ ( x ) K ( x ) − (cid:19) dx > , σ is positive, and the semi-trivial steady state ( u ∗ ( x ) ,
0) of (2.2) is unstable.Similarly, we obtain that, under the assumptions of Lemma 5, (0 , v ∗ ( x )) is also unstable. Lemma . Suppose that P ≡ K , while K and Q are linearly independent on Ω. Then the semi-trivial steady state (0 , v ∗ ( x )) of (2.2) is unstable. Lemma . Assume that P ( x ) and Q ( x ) are linearly independent on Ω, and K ( x ) ≡ αP + βQ ,with α > β >
0. Then the system (2.2) has a unique positive coexistence equilibrium ( u s , v s ) ≡ ( αP ( x ) , βQ ( x )). Proof.
A stationary solution ( u s , v s ) of system (2.2) satisfies ∇ · (cid:20) a ( x ) ∇ (cid:18) u s ( x ) P ( x ) (cid:19)(cid:21) + r ( x ) u s ( x ) (cid:18) − u s ( x ) + v s ( x ) K ( x ) (cid:19) = 0 , x ∈ Ω , ∇ · (cid:20) a ( x ) ∇ (cid:18) v s ( x ) Q ( x ) (cid:19)(cid:21) + r ( x ) v s ( x ) (cid:18) − u s ( x ) + v s ( x ) K ( x ) (cid:19) = 0 , x ∈ Ω ,∂ ( u s /P ) ∂n = ∂ ( v s /Q ) ∂n = 0 , x ∈ ∂ Ω . (3.12)The direct substitution, due to K ( x ) ≡ αP + βQ , immediately implies that ( αP ( x ) , βQ ( x )) isa coexistence stationary solution of (3.12). To show the uniqueness, assume that ( u s , v s ) ( αP ( x ) , βQ ( x )) is a coexistence equilibrium satisfying (3.12).Adding the first two equations of (3.12), integrating over Ω and taking into account the Neumannboundary conditions, we have Z Ω r ( u s + v s ) (cid:18) − u s + v s K (cid:19) dx = 0 , (3.13)which implies Z Ω rK (cid:18) − u s + v s K (cid:19) dx = Z Ω rK (cid:18) − u s + v s K (cid:19) dx > , (3.14)unless u s + v s ≡ K . However, if u s + v s ≡ K , the function w s = u s /P should satisfy ∇ · ( a ( x ) ∇ w s ) = 0 , x ∈ Ω , ∂w s /∂n = 0 , x ∈ ∂ Ω , and thus w s is constant by the Maximum Principle [16, Theorem 3.6], which means that u s /P is constant on Ω. Similarly, v s /Q is constant on Ω. From the fact that K = αP + βQ is theonly possible representation of K as a linear combination of P, Q (otherwise, P and Q are linearlydependent), we obtain u s = αP , v s = βQ .Further, dividing the second equation of (3.12) by v s /Q and integrating over Ω, we obtain Z Ω rQ (cid:18) u s + v s K − (cid:19) dx = Z Ω a |∇ ( v s /Q ) | ( v s /Q ) dx ≥ . (3.15)Next, let u s + v s K . Consider the eigenvalue problem ∇ · (cid:20) a ∇ (cid:18) φP (cid:19)(cid:21) + rφ (cid:18) − u s + v s K (cid:19) = σφ, x ∈ Ω , ∂ ( φ/P ) ∂n = 0 , x ∈ ∂ Ω (3.16)6ts principal eigenvalue σ is given by [4] σ = sup φ =0 ,φ ∈ W , − Z Ω a |∇ ( φ/P ) | dx + Z Ω r ( x ) φ P (cid:18) − u s + v s K (cid:19) dx , Z Ω φ P dx
Consider φ ( x ) = √ αP ( x ). Since αP = K − βQ , we have σ ≥ Z Ω r ( K − βQ ) (cid:18) − u s + v s K (cid:19) dx , Z Ω αP dx However, the numerator in the right-hand side equals Z Ω rK (cid:18) − u s + v s K (cid:19) dx + β Z Ω rQ (cid:18) u s + v s K − (cid:19) dx > , from (3.14) and (3.15), thus σ >
0. However, u s is the solution of ∇ · (cid:2) a ∇ (cid:0) u s P (cid:1)(cid:3) + ru s (cid:16) − u s + v s K (cid:17) = 0 , x ∈ Ω ,∂ ( u s /P ) ∂n = 0 , x ∈ ∂ Ω (3.17)and thus is a positive principal eigenfunction of (3.16) associated with the principal eigenvalue0. Thus ( u s , v s ) = ( αP ( x ) , βQ ( x )) is the unique coexistence solution of (2.2), whenever K ( x ) = αP ( x ) + βQ ( x ) and P , Q are linearly independent.Using Lemma 4 and the same scheme as in the proof of Lemma 7, we obtain Lemma . Assume that P ( x ) /K ( x ) is constant on Ω and Q ( x ), K ( x ) are linearly independent,then the system (2.2) has no coexistence equilibrium.Under the assumptions of Lemma 8, system (2.2) has the semi-trivial equilibrium ( K, Theorem . Let P ( x ) and Q ( x ) be linearly idependent on Ω, and K ( x ) ≡ αP + βQ , where α > β >
0. Then the unique coexistence solution ( u s , v s ) ≡ ( αP ( x ) , βQ ( x )) of (2.2) is globallyasymptotically stable.Note that once the trivial equilibrium is a repeller, there is no coexistence equilibrium and oneof the two semi-trivial equilibrium solutions is unstable, the other one is globally asymptoticallystable. Using Lemmata 6 and 8, we can prove the following result. Theorem . Let P ( x ) /K ( x ) be constant, P and Q be linearly independent on Ω. Then the semi-trivial equilibrium ( K ( x ) ,
0) of (2.2) is globally asymptotically stable.7 . Influence of Diffusion Coefficients and Intrinsic Growth Rates on Competition Out-come
Further, we study the dependency of the scenario (competitive exclusion or coexistence) in thecase when K is not in a positive hull of P and Q . To this end, we assume that a and a areproportional, and r is multiplied by two different constants ∂u∂t = ∇ · (cid:20) d a ( x ) P ( x ) (cid:16) ∇ u ( t, x ) − u ( t, x ) ∇ PP (cid:17)(cid:21) + r r ( x ) u ( t, x ) (cid:18) − u ( t, x ) + v ( t, x ) K ( x ) (cid:19) ,∂v∂t = ∇ · (cid:20) d a ( x ) Q ( x ) (cid:16) ∇ v ( t, x ) − v ( t, x ) ∇ QQ (cid:17)(cid:21) + r r ( x ) v ( t, x ) (cid:18) − u ( t, x ) + v ( t, x ) K ( x ) (cid:19) ,t > , x ∈ Ω ,∂u∂n − uP ∂P∂n = ∂v∂n − vQ ∂Q∂n = 0 , x ∈ ∂ Ω ,u (0 , x ) = u ( x ) , v (0 , x ) = v ( x ) , x ∈ Ω . (4.1) Lemma . Let K , P and K , Q be linearly independent, d and r be fixed. Then, for a fixed r there is d ∗ such that for d < d ∗ , the semi-trivial equilibrium (0 , v ∗ ) is unstable. For a fixed d ,there is r ∗ such that for r > r ∗ , the semi-trivial equilibrium (0 , v ∗ ) of (4.1) is unstable. Proof.
Consider the eigenvalue problem associated with the first equation in (4.1) around (0 , v ∗ ) ∇ · (cid:20) d a ( x ) ∇ (cid:18) φ ( x ) P ( x ) (cid:19)(cid:21) + r r ( x ) φ ( x ) (cid:18) − v ∗ ( x ) K ( x ) (cid:19) = σφ ( x ) , x ∈ Ω , ∂ ( φ/P ) ∂n = 0 , x ∈ ∂ Ω . (4.2)The principal eigenvalue of (4.2) is defined as σ = sup φ =0 ,φ ∈ W , − Z Ω d a |∇ ( φ/P ) | dx + Z Ω r r ( x ) φ P (cid:18) − v ∗ K (cid:19) dx , Z Ω φ P dx, and the semi-trivial equilibrium (0 , v ∗ ) is unstable if we can find φ such that the expression in theright hand side is positive. Taking φ = √ KP and using the fact that for linearly independent K , Q M := Z Ω r ( x ) K ( x ) (cid:18) − v ∗ K (cid:19) dx > − Z Ω d a |∇ ( φ/P ) | dx + Z Ω r r ( x ) φ P (cid:18) − v ∗ K (cid:19) dx = − Z Ω d a |∇ ( p K/P ) | dx + r M > d < d ∗ := r M Z Ω a |∇ ( p K/P ) | dx − or r > r ∗ := d M Z Ω a |∇ ( p K/P ) | dx, which concludes the proof. 8imilarly, for fixed r and d , for any fixed r we can find d ∗ and for any fixed d there is r ∗ such that for d < d ∗ or r > r ∗ , respectively, the equilibrium ( u ∗ ,
0) is unstable, and the secondspecies survives. Thus, by either slowing its dispersal speed or increasing its intrinsic growth rate,the species can provide its survival, unless the other chooses the optimal strategy proportional to K . However, from Lemma 9 we cannot conclude that for small d i there is coexistence, as well asfor large r i , since the semi-trivial solutions depend on these constants.Next, let us consider the cases when both populations have the same diffusion strategy with P ( x ) ≡ Q ( x ), which is linearly independent of K ( x ). Lemma . Let a ( x ) = a ( x ) = a ( x ), P ( x ) ≡ Q ( x ) satisfy ∇ · (cid:20) a ∇ (cid:18) KP (cid:19)(cid:21) , x ∈ Ω , (4.3) r = r = 1, d < d . Then there is no coexistence equilibrium of (2.2). Proof.
We assume that there is a coexistence equilibrium ( u s , v s ). Following the proof of Lemma 7,consider the eigenvalue problems ∇ · (cid:20) d a ( x ) ∇ (cid:18) φ ( x ) P ( x ) (cid:19)(cid:21) + rφ ( x ) (cid:18) − u s + v s K (cid:19) = σφ ( x ) , x ∈ Ω , ∂ ( φ/P ) ∂n = 0 , x ∈ ∂ Ω (4.4)and ∇ · (cid:20) d a ( x ) ∇ (cid:18) ψ ( x ) P ( x ) (cid:19)(cid:21) + rψ ( x ) (cid:18) − u s + v s K (cid:19) = σψ ( x ) , x ∈ Ω , ∂ ( ψ/P ) ∂n = 0 , x ∈ ∂ Ω . (4.5)The principal eigenvalue ˜ σ of (4.5) is defined as˜ σ = sup ψ =0 ,ψ ∈ W , − Z Ω d a |∇ ( ψ/P ) | dx + Z Ω r ψ P (cid:18) − u s + v s K (cid:19) dx , Z Ω ψ P dx. (4.6)However, since ( u s , v s ) is an equilibrium solution, the function v s satisfies ∇ · h d a ∇ (cid:16) v s P (cid:17)i + rv s (cid:18) − u s + v s K (cid:19) = 0 , x ∈ Ω , ∂ ( v s /P ) ∂n = 0 , x ∈ ∂ Ωand is consequently a positive principal eigenfunction of (4.5) corresponding to the principal eigen-value ˜ σ = 0. According to (4.6), − Z Ω d a |∇ ( v s /P ) | dx + Z Ω r v s P (cid:18) − u s + v s K (cid:19) dx = 0 (4.7)Also, as ( u s , v s ) is an equilibrium solution, u s satisfies ∇ · h d a ∇ (cid:16) u s P (cid:17)i + ru s (cid:18) − u s + v s K (cid:19) = 0 , x ∈ Ω , ∂ ( u s /P ) ∂n = 0 , x ∈ ∂ Ω , σ of (4.4) is defined as σ = sup φ =0 ,φ ∈ W , − Z Ω d a |∇ ( φ/P ) | dx + Z Ω r φ P (cid:18) − u s + v s K (cid:19) dx , Z Ω φ P dx. (4.8)Substituting φ = v s we get by (4.7) − Z Ω d a |∇ ( v s /P ) | dx + Z Ω r v s P (cid:18) − u s + v s K (cid:19) dx = ( d − d ) Z Ω a |∇ ( v s /P ) | dx + − Z Ω d a |∇ ( v s /P ) | dx + Z Ω r ( x ) v s P (cid:18) − u s + v s K (cid:19) dx = ( d − d ) Z Ω a |∇ ( v s /P ) | dx > , unless v s /P is constant. However, v s /P ≡ α implies u s + v s ≡ K on Ω. Substituting u s = K − αP in the first equation of (2.2) yields on Ω that0 = ∇ · (cid:20) aP (cid:18) ∇ ( K − αP ) − ( K − αP ) ∇ PP (cid:19)(cid:21) = ∇ · (cid:20) a ∇ (cid:18) KP (cid:19)(cid:21) , which contradicts to (4.3) in the assumption of the lemma. Thus σ >
0. Let us note that u s satisfies ∇ · h d a ∇ (cid:16) u s P (cid:17)i + ru s (cid:18) − u s + v s K (cid:19) = 0 , x ∈ Ω , ∂ ( u s /P ) ∂n = 0 , x ∈ ∂ Ωand thus is the positive principal eigenfunction of (4.4) corresponding to the principal eigenvalue σ = 0. The contradiction proves that there is no coexistence equilibrium. Lemma . Let P ( x ) ≡ Q ( x ) be non-proportional to K ( x ) on Ω, r = r = 1, d < d . Then thesemi-trivial equilibrium (0 , v ∗ ) of (4.1) is unstable. Proof.
The semi-trivial equilibrium (0 , v ∗ ) is a solution of the problem ∇ · (cid:20) d a ∇ (cid:18) ψ ( x ) P ( x ) (cid:19)(cid:21) + rψ ( x ) (cid:18) − v ∗ K (cid:19) = σψ ( x ) , x ∈ Ω , ∂ ( ψ/P ) ∂n = 0 , x ∈ ∂ Ω (4.9)and thus v ∗ is a positive principal eigenfunction corresponding to the zero eigenvalue of the problem ∇ · (cid:20) d a ∇ (cid:18) v ∗ ( x ) P ( x ) (cid:19)(cid:21) + rv ∗ ( x ) (cid:18) − v ∗ K (cid:19) = 0 , x ∈ Ω , ∂ ( v ∗ /P ) ∂n = 0 , x ∈ ∂ Ω . (4.10)Integrating over Ω and using the boundary conditions, we obtain − Z Ω d a |∇ ( v ∗ /P ) | dx + Z Ω r ( x ) ( v ∗ ) P (cid:18) − v ∗ K (cid:19) dx = 0 . (4.11)10o explore local stability, we notice that the linearization at (0 , v ∗ ) has the form (see, for example,[11]) ∂u ( t, x ) ∂t = ∇ · (cid:20) d a ( x ) ∇ (cid:18) u ( t, x ) P ( x ) (cid:19)(cid:21) + r ( x ) u ( t, x ) (cid:18) − v ∗ ( x ) K ( x ) (cid:19) ,∂v ( t, x ) ∂t = ∇ · h d a ( x ) ∇ (cid:16) v ( t,x ) P ( x ) (cid:17)i + r ( x ) v ( t, x ) (cid:18) − v ∗ ( x ) K ( x ) (cid:19) − r ( x ) v ∗ ( x ) u ( t,x ) K ( x ) ,t > , x ∈ Ω ,∂ ( u/P ) ∂n = ∂ ( v/P ) ∂n = 0 , x ∈ ∂ Ωand study the associated eigenvalue problems ∇ · (cid:20) d a ( x ) ∇ (cid:18) φ ( x ) P ( x ) (cid:19)(cid:21) + r ( x ) φ ( x ) (cid:18) − v ∗ ( x ) K ( x ) (cid:19) = σφ ( x ) , x ∈ Ω ,∂ ( φ/P ) ∂n = 0 , x ∈ ∂ Ω , (4.12) ∇ · (cid:20) d a ( x ) ∇ (cid:18) ψ ( x ) P ( x ) (cid:19)(cid:21) + r ( x ) ψ ( x ) (cid:18) − v ∗ ( x ) K ( x ) (cid:19) − r ( x ) v ∗ ( x ) φ ( x ) K ( x ) = σψ ( x ) , x ∈ Ω ,∂ ( ψ/P ) ∂n = 0 , x ∈ ∂ Ω . (4.13)The principal eigenvalue σ of (4.12) satisfies [4] σ = sup φ =0 ,φ ∈ W , − Z Ω d a |∇ ( φ/P ) | dx + Z Ω r φ P (cid:18) − v ∗ K (cid:19) dx , Z Ω φ P dx.
Substituting φ = v ∗ , we obtain using (4.11) σ ≥ − Z Ω d a |∇ ( v ∗ /P ) | dx + Z Ω r ( x ) ( v ∗ ) P (cid:18) − v ∗ K (cid:19) dx , Z Ω ( v ∗ ) P dx.
However, − Z Ω d a |∇ ( v ∗ /P ) | dx + Z Ω r ( v ∗ ) P (cid:18) − v ∗ K (cid:19) dx = ( d − d ) Z Ω a |∇ ( v ∗ /P ) | dx + − Z Ω d a |∇ ( v ∗ /P ) | dx + Z Ω r ( v ∗ ) P (cid:18) − v ∗ K (cid:19) dx = ( d − d ) Z Ω a |∇ ( v ∗ /P ) | dx + 0 > , as v ∗ /P is non-constant. In fact, assuming constant v ∗ /P ≡ α , we obtain from (4.10) that rv ∗ ( x ) (cid:18) − v ∗ K (cid:19) = 0 for any x ∈ Ω, or v ∗ ≡ K ≡ αP , which contradicts to the assumptionof the lemma that P is not proportional to K on Ω.Therefore the principal eigenvalue of the linearized problem is positive, which implies that theequilibrium (0 , v ∗ ) is unstable and concludes the proof.11 emark . The assumption that P is not proportional to K on Ω is a particular case of assumption(4.3). If we assume a constant a ( x ) in Ω, condition (4.3) means that the function K ( x ) /P ( x ) isnot a harmonic function on Ω, compared to being non-constant. Theorem . Let P ( x ) ≡ Q ( x ) be non-proportional to K ( x ) on Ω, r = r = 1, d < d . Then thesemi-trivial equilibrium ( u ∗ ( x ) ,
0) of (4.1) is globally asymptotically stable.
Remark . Theorem 3 generalizes the results of [9] to a more general type of diffusion in the caseof two species. Let us also note that, in the absence of diffusion, the solution of each single-speciesequation in (4.1) is K . The higher the diffusion is, the more the stationary solution deviates from K . Similarly, the following result is obtained. Theorem . Let P ( x ) ≡ Q ( x ) be non-proportional to K ( x ) on Ω, r > r , d = d = 1. Then thesemi-trivial equilibrium ( u ∗ ( x ) ,
0) of (4.1) is globally asymptotically stable.
5. Discussion
Our attempt to find the type of dispersal which includes previously known models as specialcases, initiates the following question: what is the flexibility of strategies that can lead to theideal free distribution as a stationary solution? For example, in [2, 3, 10, 11, 12, 13] in the term∆( u/P ) the dispersal strategy P was usually chosen as P ≡ K guaranteeing that K is a (globallystable) positive solution of the equation. However, if P ( x ) ≡ K ( x ) /h ( x ), where h is any harmonicfunction on Ω, K is still a solution. The boundary conditions give that the normal derivative of h on the boundary vanishes which by the Maximum Principle reduces all acceptable strategies to P ( x ) ≡ αK ( x ), α >
0. However, the constant α can be taken as a part of the diffusion coefficient,so the strategy P ≡ K is to some extent a unique optimal strategy.The results of Section 4 extend the findings of [9] to a more general type of diffusion and outlinethe coupling of the two parameters involved in the system: diffusion coefficients and intrinsicgrowth rates. The system of two equations, with the same intrinsic growth rates and α timessmaller diffusion coefficient in the first equation has the same stationary semi-trivial or coexistencesolution as the system with the same diffusion coefficient and α times larger intrinsic growth ratein the first equation. Combining this idea with the eigenvalue technique developed in [4] allows tolook at the results of [9] from a different perspective: the most productive type survives with thesame rate of dispersal, all other parameters being the same, which is certainly biologically feasible.Some of previously obtained results [2] can readily be extended to the model generalizing (2.2)to the case of two different carrying capacities ∂u∂t = ∇ · (cid:20) a ( x ) P ( x ) (cid:16) ∇ u ( t, x ) − u ( t, x ) ∇ PP (cid:17)(cid:21) + r ( x ) u ( t, x ) (cid:18) − u ( t, x ) + v ( t, x ) K ( x ) (cid:19) ,∂v∂t = ∇ · (cid:20) a ( x ) Q ( x ) (cid:16) ∇ v ( t, x ) − v ( t, x ) ∇ QQ (cid:17)(cid:21) + r ( x ) v ( t, x ) (cid:18) − u ( t, x ) + v ( t, x ) K ( x ) (cid:19) ,t > , x ∈ Ω ,∂u∂n − uP ∂P∂n = ∂v∂n − vQ ∂Q∂n = 0 , x ∈ ∂ Ω ,u (0 , x ) = u ( x ) , v (0 , x ) = v ( x ) , x ∈ Ω ,
12t least in the part of the competitive exclusion of v when, for example, P ≡ αK , r /r is constantand K ≥ K on Ω.The results obtained in the present paper strongly rely on the theory of monotone dynamicalsystems [20], and are not applicable to the case of more than two competing species. The case ofthree species for a particular diffusion strategy was considered in [14]. It would be interesting toincorporate the approach of the present paper to dispersal with consideration of more than two(and probably three) competing species, as well as patchy environment [8].In spite of the variety of diffusion models described in the present paper, they are based on thesame hypotheses:1. The domain is in some sense isolated, there is no flux across the boundary, which correspondsto the Neumann boundary conditions. This describes a closed ecosystem, all the change issubject to spatially-dependent growth laws.2. The initial-boundary value problem is designed in such a way that any nontrivial initialconditions lead to a positive and bounded solution.3. Each population diffuses on its own, i.e. the corresponding diffusion terms do not include thesecond species.The third hypothesis was challenged in [19], where it was suggested to consider the diffusion of u of the form ∆( u ( d + f ( v ))), with ∆( v ( d + f ( u ))) for the second species. If f i ≡ P , Q , these two models coincide; otherwise, they are independent. Theapproach of [19] was further developed in [17], with the homogeneous Dirichlet boundary conditions.As justified in [17], a solution is not necessarily positive, generally, the origin is a repeller: withthe appropriate design of cross-diffusion, either u or v , or both, can move out of the considereddomain. Compared to [17, Propositions 2.3], with conditions including an unknown semi-trivialsolution, Theorem 2 includes an explicit stability test; moreover, while [17, Propositions 2.3] dealswith local asymptotical stability, in the present paper we analyze the global behaviour. This canbe illustrated with [17, Propositions 4.3]: convergence to a coexistence equilibrium is stipulated bycertain initial conditions, compared to the unconditional result of Theorem 1. One of the differencesis that [17, Propositions 4.3] assumes non-symmetric interaction (growth) type, compared to thesymmetric case in the present paper. The results of [17, Propositions 4.3] can be compared to [2],where different carrying capacities for u and v were considered, and also coexistence was observed.However, the results of [17], where different diffusion type and boundary conditions were investi-gated, outline the same relations as the conclusions of Section 4: the detrimental role of self-diffusionrates and higher growth rates being a positive factor.The conclusions on cross-diffusion (see [17, Remark 4.3]) are not applicable to the model con-sidered in the present paper. It would be interesting to combine the ideas of [19] on cross-diffusionwith the above hypotheses and the general type of self-diffusion, exploring cross-interactions notonly in the growth but also in the diffusion part.
6. Acknowledgment
The authors are grateful to anonymous reviewers for their valuable comments that significantlycontributed to the presentation of the paper, to L. Korobenko for her constructive suggestions onthe previous versions of the manuscript, to Prof. Cosner for fruitful discussions during ICMA-Vmeeting in 2015. The research was supported by NSERC grant RGPIN-2015-05976.13 eferences [1] I. Averill, Y. Lou and D. Munther, On several conjectures from evolution of dispersal,
J. Biol.Dyn. (2012), 117–130.[2] E. Braverman, Md. Kamrujjaman and L. Korobenko, Competitive spatially distributed pop-ulation dynamics models: does diversity in diffusion strategies promote coexistence? Math.Biosci. (2015), 63-73.[3] E. Braverman and L. Braverman, Optimal harvesting of diffusive models in a non-homogeneousenvironment,
Nonlin. Anal. Theory Meth. Appl. (2009), e2173–e2181.[4] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-diffusion Equations, Wiley Seriesin Mathematical and Computational Biology , John Wiley & Sons, Chichester, 2003.[5] R. S. Cantrell, C. Cosner and Y. Lou, Movement toward better environments and the evolutionof rapid diffusion,
Math. Biosci. (2006), 199–214.[6] R. S. Cantrell, C. Cosner and Y. Lou, Approximating the ideal free distribution via reaction-diffusion-advection equations,
J. Differential Equations (2008), 3687-3703.[7] R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and the ideal free distribution,
Math. Biosci. Eng. (2010), 17–36.[8] R. S. Cantrell, C. Cosner and Y. Lou, Evolutionary stability of ideal free dispersal strategiesin patchy environments, J. Math. Biol. (2012), 943-965.[9] J. Dockery, V. Hutson, K. Mischaikow and M. Pernarowski, The evolution of slow dispersalrates: a reaction diffusion model, J. Math. Biol. (1998), 61–83.[10] L. Korobenko and E. Braverman, A logistic model with a carrying capacity driven diffusion, Can. Appl. Math. Quart. (2009), 85–100.[11] L. Korobenko and E. Braverman, On logistic models with a carrying capacity dependentdiffusion: stability of equilibria and coexistence with a regularly diffusing population, NonlinearAnal. B: Real World Appl. (2012), 2648–2658.[12] L. Korobenko and E. Braverman, On evolutionary stability of carrying capacity driven disper-sal in competition with regularly diffusing populations, J. Math. Biol. (2014), 1181–1206.[13] L. Korobenko, Md. Kamrujjaman and E. Braverman, Persistence and extinction in spatialmodels with a carrying capacity driven diffusion and harvesting, J. Math. Anal. Appl.
Discrete Contin. Dyn.Syst. (2012), 3099–3131.[15] X. Q. He and W. M. Ni, The effects of diffusion and spatial variation in Lotka-Volterracompetition-diffusion system I: Heterogeneity vs. homogeneity, J. Differential Equations (2013), 528-546. 1416] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, secondedition, Springer-Verlag, Berlin, 1983.[17] Y. Jia, J. Wu and H. K. Xu, Positive solutions of a Lotka-Volterra competition model withcross-diffusion,
Comput. Math. Appl. (2014), 1220-1228.[18] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum, New York, 1992.[19] N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species. J.Theoret. Biol.79