# Distinct Spreading Speeds in a Lotka–Volterra Cooperative System

Authors: Shuxia Pan
Journal: Journal of Mathematics
 (2020)

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## Abstract

This paper deals with the spreading speeds in the classical Lotka–Volterra cooperative system, of which the bounds have been studied earlier. By introducing an auxiliary cooperative system and constructing an upper solution, we obtain a sufficient condition to confirm two distinct spreading speeds of unknown functions. Due to the different average moving speeds of different level sets, we find the existence of propagation terraces in such a cooperative system with constant coefficients. We also present some numerical results to illustrate our results.


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## Body

## 1. Introduction

There are many examples where the interaction of two or more species is to the advantage of all, which is described by cooperative or mutualism models in population dynamics. Considering the limited carrying capacities for both species, one basic cooperative model is the following Lotka–Volterra type system:(1)dN1dt=r1N11−N1+b1N2,dN2dt=r2N21−N2+b2N1,in which all the parameters are positive and N1,N2 represent the densities of two cooperative species. We refer to Murray (Section 3.6 [1]) for the dynamics of (1). Introducing the spatial movement of individuals, one reaction-diffusion model is given by [2](2)∂u1x,t∂t=d1Δu1x,t+r1u1x,t1−u1x,t+b1u2x,t,∂u2x,t∂t=d2Δu2x,t+r2u2x,t1−u2x,t+b2u1x,t,in which x∈ℝ,t>0,Δ=∂2/∂x2 and all the parameters are positive. If b1b2<1, then (2) admits a positive steady state that is spatially homogeneous and defined by(3)E=e1,e2=1+b11−b1b2,1+b21−b1b2.From the viewpoint of population dynamics, an important problem is to formulate the dynamics of (2) when the initial values satisfy proper conditions. In particular, assuming that u1x,0,u2x,0 have nonempty compact supports, Li et al. [2] and Lin [3] studied the corresponding initial value problem of (2) by the following propagation threshold [4].Definition 1.
Assume thatux,t is a nonnegative function for all x∈ℝ,t>0. Then cu is called the spreading speed of ux,t if(a)
limt⟶∞supx>cu+εtux,t>0 for any given ϵ>0(b)
liminft⟶∞infx<cu−εtux,t>0 for any given ϵ∈0,cu
More precisely, Li et al. (Example 4.1 in [2]) estimated that(4)2diri≤cui≤2diriei,i=1,2,and some other bounds when d1r1>d2r2e2. Moreover, Lin (Theorem 3.1 in [3]) proved that if d1r1>d2r2e2, then cu2≥2d2r21+b2,cu1=2d1r1. The purpose of this paper is to present some results such that cu2=2d2r21+b2 if d1r1>d2r2e2.
By Lin (Theorem 3.1 in [3]), it suffices to study the upper bounds of cu2 if d1r1>d2r2e2. For this purpose, we introduce an auxiliary cooperative system that admits weaker irreducible property. The classical results in Weinberger et al. [5] imply the existence of constant speed of asymptotic spreading. However, this system is not subhomogeneous, and we cannot use some classical results of monotone semiflows to present the speed. In this paper, we present some sufficient conditions such that cu2=2d2r21+b2, which is based on proper upper and lower solutions and comparison principle. Furthermore, we give some numerical results to explore both the role of linear part and nonlinear part in the reaction term.


## 2. Main Results

Consider the initial value problem(5)∂u1x,t∂t=d1Δu1x,t+r1u1x,t1−u1x,t+b1u2x,t,∂u2x,t∂t=d2Δu2x,t+r2u2x,t1−u2x,t+b2u1x,t,u1x,0=ψ1x,u2x,0=ψ2x,in which x∈ℝ,t>0, positive parameters satisfy(6)0<b1b2<1,d1r1>d2r2e2.The initial conditionsψ1,ψ2 are bounded and uniformly continuous functions admitting nonempty supports such that(7)0≤ψix≤ei,x∈ℝ,i=1,2.By the basic theory of reaction-diffusion systems [6, 7], we have the following conclusion.Lemma 1.
Equation (5) has a global classical solution such that(8)0≤uix,t≤ei,x∈ℝ,t>0,i=1,2.Study the following auxiliary system(9)∂U1x,t∂t=d1ΔU1x,t+r1U1x,t+1−U1x,t+b1U2x,t,∂U2x,t∂t=d2ΔU2x,t+r2U2x,t1+b2−U2x,t+b2U1x,t,U1x,0=ϕ1x,U2x,0=ϕ2x,where x∈ℝ,t>0, and(10)ϕ1x=max0,ψ1x−1,ϕ2x=ψ2x,x∈ℝ.The following existence of classical solutions of (9) is true by the theory in [6, 7].Lemma 2.
Equation (9) has a global classical solution such that(11)0≤U1x,t≤e1−1,0≤U2x,t≤e2,x∈ℝ,t>0.Moreover, the comparison principle implies that(12)maxu1x,t−1,0≤U1x,t,u2x,t≤U2x,t,x∈ℝ,t>0.At the steady state0,0, the linearizing system of (9) is(13)∂U1x,t∂t=d1ΔU1x,t−r1U1x,t+r1b1U2x,t,∂U2x,t∂t=d2ΔU2x,t+r21+b2U2x,t.Evidently, (13) is not irreducible, but the hair-trigger effect in (9) occurs once ϕ2x admits nonempty support even if ϕ1x=0,x∈ℝ. The phenomenon is similar to that in competition-exclusion process in diffusive competitive systems (see Weinberger et al. [5]). Again by the comparison principle and the results stated in Section 1 (Li et al. (Example 4.1 in [2]) and Lin (Theorem 3.1 in [3])), we have the following results on spreading speed.Lemma 3.
cu2=2d2r21+b2 if cU2=2d2r21+b2.By Lemma3, it suffices to give some sufficient conditions such that cU2=2d2r21+b2. Note that (9) is cooperative but not subhomogeneous; then Weinberger et al. [5] implies that(14)cU2≥2d2r21+b2,cU1≥2d2r21+b2,which is also clear by utilizing the comparison principle and studying the following auxiliary equation [4]:(15)∂wx,t∂t=d2Δwx,t+r2wx,t1+b2−wx,t,x∈ℝ,t>0.Now, we prove thatcU2=2d2r21+b2 by constructing upper and lower solutions. For the purpose, we introduce some notations. Define(16)c∗=2d2r21+b2,λ=r21+b2d2.We shall prove the following result on spreading speed.Theorem 1.
Assume thatsupx∈ℝϕ1x+ϕ2xeλx<∞. If(17)2r21+b2≥d1r21+b2d2+r1b1b2−1,then cU1=cU2=2d2r21+b2 such that cu1=2d1r1>cu2=2d2r21+b2.Proof.
Define(18)U¯1x,t=mineλx+c∗t+x0,eλ−x+c∗t+x0,e1−1,U¯2x,t=minb2eλx+c∗t+x0,b2eλx+c∗t+x0,e2,in which x0>0 is a constant such that(19)U¯1x,0≥ϕ1x,U¯2x,0≥ϕ2x,x∈ℝ.
IfU¯1x,t,U¯2x,t is an upper solution of (9), then the result is true by the comparison principle of cooperative systems. We now prove the definition of upper solutions.
We first prove that(20)∂U¯2x,t∂t≥d2ΔU¯2x,t+r2U¯2x,t1+b2−U¯2x,t+b2U¯1x,t,when U¯2x,t is differentiable, which is clear if U¯2x,0=e2. When U¯2x,t=b2eλx+c∗t+x0 is differentiable, then(21)∂U¯2x,t∂t=d2ΔU¯2x,t+r2U¯2x,t1+b2≥d2ΔU¯2x,t+r2U¯2x,t1+b2−U¯2x,t+b2U¯1x,t,d2ΔU¯2x,t−∂U¯2x,t∂t+r2U¯2x,t1+b2=b2eλx+c∗t+x0d2λ2−c∗λ+r21+b2=0,such that (20) is true by the definitions of λ,c∗. Similarly, if U¯2x,t=b2eλ−x+c∗t+x0 is differentiable, then we can obtain the same inequality (20).
OnU1, we need to prove that(22)∂U¯1x,t∂t≥d1ΔU¯1x,t+r1U¯1x,t+1−U¯1x,t+b1U¯2x,t,when U¯1x,t is differentiable, which is true if U¯1x,t=e1−1 since(23)r1U¯1x,t+1−U¯1x,t+b1U¯2x,t≤r1e1−e1−1+b1e2=r1e11−e1+b1e2=0.
WhenU¯1x,t=eλx+c∗t+x0 is differentiable, then(24)∂U¯1x,t∂t−d1ΔU¯1x,t=c∗λ−d1λ2eλx+c∗t+x0,=2r21+b2−d1r21+b2d2eλx+c∗t+x0,r1U¯1x,t+1−U¯1x,t+b1U¯2x,t≤r1b1b2−1U¯1x,t+1eλx+c∗t+x0,≤r1b1b2−1eλx+c∗t+x0.
Byb1b2<1, then (22) is true if (17) holds. Similarly, if U¯1x,t=eλ−x+c∗t+x0 is differentiable, then (22) holds. The proof is complete. □
Once the spreading speed is confirmed, we may obtain the convergence results in different moving intervals as time goes to infinity by the monotonicity and Fatou lemma, which is similar to that in the proof of Lin (Theorem 3.1 in [3]).Theorem 2.
Assume that Theorem1 holds. Then for any given(25)2ϵ∈0,2d1r1−2d2r21+b2,we have cu1=2d1r1,cu2=2d2r21+b2 and(26)limsupt⟶∞supcu2+εt<x<cu1−εtu1x,t=liminft⟶∞infcu2+εt<x<cu1−εtu1x,t=1,limsupt⟶∞supx<cu2−εtuix,t=liminft⟶∞infx<cu2−εtuix,t=ei,i=1,2,limsupt⟶∞supx>cu1+εtu1x,t=0,limsupt⟶∞supx>cu2+εtu2x,t=0.


## 3. Numerical Simulation

In this section, we simulate several cases of spreading speeds of (2). Define(27)Lλut=infx:ux,t=λ,which is a special level set of ux,t. If t1>t2>0, then Lλut1−Lλut2/t2−t1 may formulate the average movement speed of the level sets in time interval t2,t1, of which the limit may further describe the spreading speed of ux,t as t1−t2 goes to infinity.Example 1.
We take(28)d1=d2=1,r1=1,r2=0.3,b1=0.2,b2=0.3,and initial value(29)ψ1x=ψ2x=cosx,x<π2,ψ1x=ψ2x=0,x≥π2.
Evidently, (17) is true. By our theory, we have cu1=2,cu2=1.2. From Figures 1 and 2, we see that two species invade the habitat almost at two distinct constant speeds. Furthermore, L0.1u1tandL1.1u1t have different moving speeds, and the movement speed of L0.1u1t (L1.1u1t and L1.1u2t) is close to cu1 (cu2).Figure 1
Simulation ofu1x,t,u2x,t defined by (2) with (28) and (29).Figure 2
Spatial plots ofu1x,t (green line) and u2x,t (blue line) at t=100,150 defined by (2) with (28) and (29).Example 2.
We take(30)d1=d2=1,r1=1,r2=0.3,b1=b2=0.2,and initial value(31)ψ1x=1,x∈ℝ,ψ2x=cosx,x<π2,ψ2x=0,x≥π2.
Evidently, (17) is true. By our theory, we have cu1−1=2andcu2=1.2. Although u1x,0 is not compactly supported, the spreading speed of u2 is close to our results from Figures 3 and 4. Moreover, the movement speed of both L1.1u1t and L1.1u2t is close to cu2.Figure 3
Simulation ofu1x,t,u2x,t defined by (2) with (28) and (29).Figure 4
Spatial plots ofu1x,t (green line) and u2x,t (blue line) at t=100,150 defined by (2) with (28) and (29).Example 3.
We take(32)d1=1,d2=13,r1=r2=1,b1=b2=0.2,and initial value(33)ψ1x=ψ2x=cosx,x<π2,ψ1x=ψ2x=0,x≥π2.
Evidently, (17) is not true. If our theory were true, then we have(34)cu1=2,cu2=2105≈1.264.
From numerical results (Figures5 and 6), the threshold may be true.Figure 5
Simulation ofu1x,t,u2x,t defined by (2) with (32) and (33).Figure 6
Spatial plots ofu1x,t (green line) and u2x,t (blue line) at t=100,150 defined by (2) with (32) and (33).


## 4. Discussion

The cooperative systems have special dynamical features, for example, many cooperative systems do not admit nontrivial periodic solutions [8]. When the propagation dynamics is concerned, much attention has been paid to monotone semiflows (see Fang et al. [9], Liang and Zhao [10], Lui [11], and Weinberger et al. [5]). If a cooperative system is reducible at some invadable steady state (unstable steady state), then it is possible that there are several different spreading speeds, which at least has been observed in (2) (see Li et al. [2] and Lin [3]). However, it remains open to present complete conclusion of spreading speed of (2), and it is a challenging question to further develop the propagation theory of monotone semiflows that are not subhomogeneous and irreducible.For scalar equations with constant coefficients, the propagation dynamics has been widely explored. In particular, it has been observed a unique spreading speed in both monostable case and bistable case [4, 12] if the solutions could spread. When the coefficients depend on spatial or temporal variables, the propagation dynamics may be more complex [13]. In particular, different propagation terraces occur due to the different movement speeds of different level sets [14]. Although the propagation terraces do not occur in classical monostable or bistable scalar reaction-diffusion equations with constant coefficients, we can observe different terraces in (2). To further show different spreading speeds of different terraces is also an interesting question.



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