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Asymptotic spreading speeds for a predator-prey system with two predators and one prey
Arnaud Ducrot, Thomas Giletti, Jong-Shenq Guo, Masahiko Shimojo
To cite this version:
Arnaud Ducrot, Thomas Giletti, Jong-Shenq Guo, Masahiko Shimojo. Asymptotic spreading speeds for a predator-prey system with two predators and one prey. Nonlinearity, IOP Publishing, 2020, 34 (2), pp.669-704. �10.1088/1361-6544/abd289�. �hal-03264552�
SYSTEM WITH TWO PREDATORS AND ONE PREY
ARNAUD DUCROT, THOMAS GILETTI, JONG-SHENQ GUO, AND MASAHIKO SHIMOJO
Abstract. This paper investigates the large time behaviour of a three species reaction-diffusion system, modelling the spatial invasion of two predators feeding on a single prey species. In addition to the competition for food, the two preda- tors exhibit competitive interactions and, under some parameter condition, they can also be considered as two mutants. When mutations occur in the predator popula- tions, the spatial spread of invasion takes place at a definite speed, identical for both mutants. When the two predators are not coupled through mutation, the spread- ing behaviour exhibits a more complex propagating pattern, including multiple layers with different speeds. In addition, some parameter conditions reveal situations where a nonlocal pulling phenomenon occurs and in particular where the spreading speed is not linearly determined.
1. Introduction
In this work we study the large time behaviour for the following three-species predator-prey system
ut=d1uxx+uF(u, v, w) +µ(v−u) in (0,+∞)×R, (1.1)
vt=d2vxx+vG(u, v, w) +µ(u−v) in (0,+∞)×R, (1.2)
wt=d3wxx+wH(u, v, w) in (0,+∞)×R, (1.3)
wherein the functions F,G and H are respectively given by F(u, v, w) := r1(−1−u−kv+aw), G(u, v, w) := r2(−1−hu−v+aw), H(u, v, w) := r3(1−bu−bv−w).
Hereu=u(t, x) andv =v(t, x) denote the densities of two predators and w=w(t, x) corresponds to the density of the single prey. The parameters di, ri in system (1.1)- (1.3) are all positive and respectively stand for the motility and per capita growth rate of each species. The parameters h > 0 and k > 0 represent the competition between the two predators whilea >0 and b >0 describe the predator-prey interactions. The parameterµ is assumed to be nonnegative.
Date: July 4, 2020. Corresponding author: J.-S. Guo.
2010 Mathematics Subject Classification. 35K45, 35K57, 35K55, 92D25.
Key words and phrases: predator-prey system, spreading speed, competition, mutation, nonlocal pulling.
1
When µ > 0 it stands as a mutation rate between speciesu and v, so that the two predators are mutant species. Therefore, one should distinguish the caseµ= 0, where uand v can be understood as completely distinct species which are competing for the same prey, from the case µ >0 where u and v are the densities of two mutants of the same predator species, both feeding on the same prey. Both situations also lead to very different large time behaviours of the solutions. We shall refer to the former as the ‘two competitors’ case, and to the latter as the ‘two mutants’ case.
To perform our study of system (1.1)-(1.3), we impose the following parameter conditions
(1.4) a >1, 0< h, k <1, 0< b < 1
2(a−1), 0≤µ≤ a−1
2 min{r1, r2}. Condition (1.4) shall be assumed throughout this paper, and as we shall see below it typically allows the possibility of co-existence of all three species.
The main goal of this paper is to study the asymptotic spreading speed(s) of the two invading predators into a prey population uniformly close to its carrying capacity.
For this purpose, we typically investigate the Cauchy problem for (1.1)-(1.3) with the initial condition
(1.5) u(0, x) =u0(x), v(0, x) =v0(x), w(0, x) =w0(x), x∈R,
where initial data u0, v0, w0 are uniformly continuous functions defined onR, both u0 and v0 have nontrivial compact supports, and
(1.6) 0≤u0, v0 ≤a−1, β≤w0 ≤1 on R.
Hereafter, we setβ := 1−2b(a−1)>0 in which the positivity follows from the third condition in (1.4). Note that by the classical theory of reaction-diffusion systems there is a unique global (in time) solution (u, v, w) of the Cauchy problem for system (1.1)- (1.3) with initial condition (1.5) for any uniformly continuous initial data (u0, v0, w0) satisfying (1.6).
The description of spatial propagation for diffusive predator-prey systems has a long history. In particular, traveling wave solutions have been exhibited in a wide range of predator-prey systems. For the systems with exactly one prey and one predator, we refer in particular to the pioneering works [9, 10]. We also refer to [18, 16, 20] and the references cited therein for more results about traveling waves in such a context.
Recently, more attention is paid to the existence of traveling waves of some three species predator-prey systems. For this, we refer the reader to, e.g., [2, 17] for a diffusive system involving two preys and one predator, and [15, 19] for a system with two predators and one prey.
While some stability results were obtained in [11], it is only recently that a more exhaustive study of the spreading properties of solutions of the Cauchy problem has been performed for systems with one prey and one predator [5, 6, 8, 22]. This naturally raises the question of propagation phenomena in more realistic systems involving a larger number of interacting species. Let us mention in particular that the spreading
speed of the predator for a three-species predator-prey system with a single predator and two preys was studied by Wu [23].
As far as mutant systems are considered, we refer to Griette and Raoul [14] for a study of traveling wave solutions. We also refer to Girardin [12] for a result on the asymptotic spreading speed of solutions of the Cauchy problem and to Morris et al [21] for discussion about the linear determinacy of the spreading speed. However, those earlier works were concerned with systems coupling mutations and competitive interactions between the mutants. As far as we know, our work is the first to investigate the spreading properties in a situation involving both mutations and prey-predator interactions.
One important difficulty to be overcome in our analysis of the spreading properties for system (1.1)-(1.3) with (1.5) is the lack of comparison principle. In addition, as described below, the ‘two-competitors’ system admits different semi-trivial equilibria that generate complex propagating behaviour, including the development of multiple propagating layers with different speeds. These difficulties are overcome by using refined dynamical system arguments and by carefully investigating suitable omega- limit solutions in various and adapted moving frames.
The rest of this paper is organized as follows. First, Section 2 is devoted to the description of the main results obtained in this paper. This comes after introducing some notation to be used throughout this paper. Then we start in Section 3 with some preliminaries. These preliminaries include a priori estimates on solutions and the computation of the principal eigenvalues of some linearized systems which shall arise in the proof of our main results. Moreover, we establish some persistence result, Lemma 3.2, which shall be a key ingredient in our arguments, in the spirit of [6].
Then, in Section 4, we show by a Lyapunov like argument that, in the ‘two competitors case’, bounded entire in time solutions which are bounded from below by some positive constants must be identical to a steady state.
The later sections are directly concerned with the proofs of our main results de- scribed in Section 2. Section 5 mostly deals with the ‘two mutants’ case and some general upper bounds on the spreading speeds, both for the ‘two mutants’ and ‘two competitors’ cases. Next, in Section 6 we establish some of the lower bounds on the speeds in the ‘two competitors’ case. Then in Section 7 we deal with the case when both predator species have the same speed and diffusivity. Finally, in Section 8, we first derive the definite spreading speed of the faster predator in the ‘two competitors’
case. Then we exhibit a ‘nonlocal pulling’ phenomenon for the slower predator. This leaves a question on the definite spreading speed of the slower predator to be open.
2. Main results
In this section, we state the main results that shall be proved and discussed in this paper. We start by introducing some notation and important quantities before going to the description of our main spreading results.
2.1. Some notation.
2.1.1. Constant equilibria. Let us first look at the underlying ODE system. It is clear that (0,0,0) and (0,0,1) (the predator-free state) are two trivial equilibria for system (1.1)-(1.3); both are linearly unstable. For other steady states, we consider separately the ‘two competitors’ and the ‘two mutants’ cases. In both cases, we shall find that the three components ultimately persist in the large time behaviour of solutions of (1.1)- (1.3). Furthermore, in the first case, the solution converges to the unique co-existence steady state which we compute below.
Two competitors case µ = 0: It is easy to check that, when a > 1, (0,p,˜ q) and˜ (˜p,0,q) are two semi-trivial steady states of system (1.1)-(1.3), where˜
(2.1) p˜= a−1
1 +ab, q˜= b+ 1 1 +ab.
Moreover, under condition (1.4) there is a unique positive co-existence state (u∗, v∗, w∗), where
u∗ := 1−k
1−hk(aw∗−1), v∗ := 1−h
1−hk(aw∗−1), w∗ := (1−hk) +b(2−h−k)
(1−hk) +ab(2−h−k). (2.2)
Note that one has F(0,p,˜ q) =˜ r1
1 +ab(a−1)(1−k)>0, G(˜p,0,q) =˜ r2
1 +ab(a−1)(1−h)>0.
This means that the semi-trivial steady states are unstable with respect to the under- lying kinetic ODE system.
Two mutants case µ > 0: In this case, because of the coupling between the two mutants it is clear that there cannot exist a semi-trivial steady state of system (1.1)- (1.3). However, as in the case µ = 0, there typically exists a unique positive co- existence steady state (u∗µ, vµ∗, wµ∗). This can be checked proceeding as in [14] provided thatµis small enough. Note that, as discussed in [3], the mutation rate µis typically small for relevant biological situations. Since a more detailed analysis of the equilibria is not necessary to our purpose, we omit the details.
2.1.2. Linear speeds. We shall take an interest in the invasion of predators on the predator-free state. Consider the linearization of (1.1) and (1.2) around the predator- free state (0,0,1), and write it in matrix form as
( ut vt
)
=
( d1uxx d2vxx
)
+M[µ]× ( u
v )
, where
M[µ] :=
[ r1(a−1)−µ µ µ r2(a−1)−µ
] .
In the case whenµ >0, the componentsu and v are mutants of the same species and therefore they should spread simultaneously. By analogy with the scalar equation, one may want to look for an ansatz of the type
(u, v) = (p, q)e−γ(x−ct), p, q, γ >0, c ∈R.
Putting this in the above equation, one finds that (p, q) andγ must solve M[µ, γ]×
( p q
)
=cγ ( p
q )
, where
(2.3) M[µ, γ] :=
[ d1γ2+r1(a−1)−µ µ
µ d2γ2+r2(a−1)−µ ]
. This leads us to introduce
(2.4) c∗µ := min
γ>0
Λ[µ, γ]
γ >0,
such that an ansatz of the above type solves the linearized system around the predator- free state if and only if c ≥ c∗µ. Here Λ[µ, γ] denotes the unique principal eigenvalue of M[µ, γ], which exists by applying Perron-Frobenius theorem. We point out that the existence and positivity ofc∗µ are ensured by the fact that Λ[µ,0]>0 (recall that 2µ≤(a−1) min{r1, r2}) and γ 7→Λ[µ, γ] is convex.
In the case when µ = 0, the functions u and v are population densities of distinct species which may spread with different speeds. This can be seen in the fact that the matrix M[0] is no longer irreducible; one can then find semi-trivial ansatzes
(u, v) = (p,0)e−γ(x−ct), (u, v) = (0, q)e−γ(x−ct), p, q >0.
These exist if, respectively,c≥c∗u and c≥c∗v. Those speeds are explicitly defined as
(2.5) c∗u := 2√
d1r1(a−1), c∗v := 2√
d2r2(a−1).
One may check thatc∗µ converges to max{c∗u, c∗v} asµ→0. Without loss of generality, we may assume thatd1r1 ≥d2r2. Hence throughout this paper we always assume that
c∗u ≥c∗v.
In the case when µ = 0, it is then expected that the component u shall spread with speedc∗u, and the componentv at some slower speed. However, this means thatv may no longer invade in the predator-free state. This leads us to introduce
c∗∗u := 2
√ d1r1
1 +ab(a−1)(1−k) =c∗u
√1−k 1 +ab, c∗∗v := 2
√1−h
1 +abd2r2(a−1) = c∗v
√1−h 1 +ab. (2.6)
Note thatc∗∗v < c∗v and c∗∗v is the spreading speed of solutions of vt =d2vxx+r2v(−1−hp˜+a˜q),
i.e., the linear invasion speed into the semi-trivial steady state (whenµ= 0) where u and w coexist.
2.2. Our main results. We are now in a position to describe the spreading speed of solutions. In the sequel, we let (u, v, w) be a solution of the Cauchy problem for system (1.1)-(1.3) with the initial condition (1.5), under the assumption (1.6).
Our first theorem deals with the ‘two mutants’ case µ > 0 where we can describe accurately the spreading speed of the solution:
Theorem 2.1. Assume that (1.4) holds and that in addition µ > 0. Let (u0, v0) be nontrivial and compactly supported such that 0 ≤ u0, v0 ≤ a −1, and β ≤ w0 ≤ 1.
Recall also that c∗µ is defined by (2.4). Then the solution (u, v, w) enjoys the following spreading behaviour:
(i) for any c > c∗µ,
t→lim+∞ sup
|x|≥ct
{|u(t, x)|+|v(t, x)|+|1−w(t, x)|}= 0;
(ii) for any c∈[0, c∗µ), lim inf
t→+∞ inf
|x|≤ctmin{u(t, x), v(t, x),1−w(t, x)}>0.
The above result shows that both predator mutants spread with the same asymptotic speed. However, the question of the convergence to a steady state in the wake of the propagation remains open.
Let us now turn to the ‘two competitors’ case µ = 0. This situation turns out to be more complicated, because unlike in the ‘two mutants’ case the subsystem (1.1) and (1.2) with constantw has no cooperative structure in the neighborhood of (0,0).
Moreover, both competitors may spread with different spreading speeds, so that ul- timately one may observe three zones: ahead of the propagation in the predator-free state, after the propagation of the first predator but before arrival of the second, and finally a co-existence state where all three species persist.
The next theorem provides some estimates on the spreading speeds of both species.
Theorem 2.2. Assume that (1.4)holds and that in additionµ= 0. Let both u0 andv0 be nontrivial and compactly supported such that 0≤ u0, v0 ≤a−1, and β ≤w0 ≤1.
Recall also thatc∗u andc∗v are defined in (2.5). Then the solution (u, v, w) satisfies, for any c > c∗u,
t→lim+∞ sup
|x|≥ct
u(t, x) = 0, and, for any c > c∗v,
t→+∞lim sup
|x|≥ct
v(t, x) = 0.
Moreover, for any c >max{c∗u, c∗v},
t→lim+∞ sup
|x|≥ct
|1−w(t, x)|= 0.
In particular, this result states that u and v spread at most, respectively, with speeds c∗u and c∗v. Now the question is whether c∗u and c∗v are indeed the spreading speed ofuand v. To answer this point, we consider separately the cases whenc∗v =c∗u and c∗v < c∗u.
Theorem 2.3. Assume that (1.4) holds and that in addition µ= 0 and c∗v = c∗u. Let both u0 and v0 be nontrivial and compactly supported such that 0 ≤ u0, v0 ≤ a−1, and β ≤ w0 ≤ 1. Recall also that c∗∗u and c∗∗v are defined in (2.6). Then the solution (u, v, w) enjoys the following spreading behaviour:
(i) for each c∈(0, c∗v) one has lim inf
t→+∞ inf
|x|≤ct(u+v) (t, x)>0;
(ii) the functions u and v separately satisfy lim inf
t→+∞ inf
|x|≤ct
u(t, x)>0, ∀c∈(0, c∗∗u ), lim inf
t→+∞ inf
|x|≤ctv(t, x)>0, ∀c∈(0, c∗∗v );
(iii) finally, for each 0< c <min(c∗∗v , c∗∗u ),
t→lim+∞ sup
|x|≤ctk(u, v, w)(t, x)−(u∗, v∗, w∗)k= 0,
wherein(u∗, v∗, w∗)is the constant equilibrium defined in (2.2)andk·kdenotes any norm in R3.
In the equi-diffusion case, the previous result can be strengthened as follows. In the sequel, we setκ:= (1−k)/(1−h).
Theorem 2.4.Under the same assumptions as Theorem 2.2, suppose also thatw0 ≡1, d1 =d2 and r1 =r2, so that in particular c∗u =c∗v. Then, for any c∈[0, c∗u),
lim inf
t→+∞ inf
|x|≤ctu(t, x)>0, if u0 ≥κv0; while, for any c∈[0, c∗v), if u0 ≤κv0 we have
lim inf
t→+∞ inf
|x|≤ctv(t, x)>0.
In each case, we also have
lim sup
t→+∞ sup
|x|≤ct
w(t, x)<1.
We now turn to the case whenc∗v < c∗u, where the following result answers positively that c∗u is indeed the spreading speed of u.
Theorem 2.5. Assume that (1.4) holds and that in addition µ= 0 and c∗v < c∗u. Let bothu0 and v0 be nontrivial and compactly supported such that 0≤u0, v0 ≤a−1, and β≤w0 ≤1. Then the solution (u, v, w) enjoys the following spreading behaviour:
(i) for each c∈[0, c∗u), one has lim inf
t→+∞ inf
|x|≤ctu(t, x)>0;
(ii) for any c∗v < c1 < c2 < c∗u,
t→lim+∞ sup
c1t≤|x|≤c2t
k(u, v, w)(t, x)−(˜p,0,q)˜k= 0, wherein (˜p,0,q)˜ is the constant equilibrium defined in (2.1);
(iii) finally, for each 0< c < c∗∗v ,
t→lim+∞ sup
|x|≤ct
k(u, v, w)(t, x)−(u∗, v∗, w∗)k= 0.
Together with Theorem 2.2, this describes accurately the spreading of the faster predator when c∗u > c∗v. However, it is still only known that v spreads at least at speed c∗∗v , and at most at speed c∗v. It remains an open question whether v has a definite spreading speed and what this spreading speed is. While one may expect that the spreading speed ofv is c∗∗v , which indeed corresponds to the linear invasion speed into the intermediate semi-trivial steady state (˜p,0,q), we can actually construct a˜ situation where the spreading speed is strictly faster than c∗∗v . This is comparable to the nonlocally pulled phenomenon which occurs in competition systems [13].
Theorem 2.6 (Nonlocal Pulling). Under the same assumptions as in Theorem 2.5, assume furthermore that
1−2ab−h >0, c∗∗v −√
(c∗∗v )2−4d2r2(aβ−1−h(a−1))> (c∗u)2−(c∗v)2 2(c∗u−c∗∗v ) . Then there exists c > c∗∗v (independent of the initial data) such that
lim inf
t→+∞ inf
|x|≤ctv(t, x)>0.
Remark 2.1. The conditions proposed above hold in particular when 1−2ab−h >0 and0< d1r1−d2r2 is sufficiently small. This ensures that there is a set of parameters which satisfy the assumptions of Theorem 2.6, thus the nonlocal pulling phenomenon does indeed occur.
3. Preliminaries
3.1. Some a priori estimates. Let (u, v, w) be a solution of the Cauchy problem for system (1.1)-(1.3) with the initial condition (1.5), under the assumption (1.6). It is clear that the solution (u, v, w) exists globally. By the (strong) comparison principle (for scalar equations), it is easy to see that u, v, w > 0 in (0,∞)×R, and w < 1 in (0,∞)×R. Then
{
ut ≤d1uxx+r1u(a−1−u) +µ(v−u), vt≤d2uxx+r2v(a−1−v) +µ(u−v).
This system is of the cooperative type and satisfies a comparison principle. It is then straightforward thatu, v < a−1 in (0,∞)×R. Moreover, by (1.6) and (1.3), another comparison principle gives that w≥β in (0,∞)×R.
We sum up the above in the following proposition.
Proposition 3.1. Assume that (u, v, w) solves (1.1)-(1.3) together with (1.5) with µ≥ 0, where 0≤u0, v0 ≤a−1, β ≤w0 ≤ 1. Then the inequalities 0≤u, v ≤a−1, β≤w≤1 also hold for all t >0.
3.2. Orbit closure sets and key persistence lemmas. In this section and for convenience, we introduce
(3.1) X0 :={(u0, v0, w0)∈(U C0(R;R))3| 0≤u0, v0 ≤a−1, β≤w0 ≤1}, where U C0(R;R) denotes the set of uniformly continuous and bounded functions fromR toR. In other words, X0 denotes the set of initial data satisfying (1.6).
According to Proposition 3.1, for any (u0, v0, w0) ∈ X0, the associated solution (u, v, w) satisfies the same inequalities for all positive times, i.e.,
0≤u, v ≤a−1, β ≤w ≤1.
We also observe the following property. Let (u, v, w) be a solution of (1.1)-(1.3) with initial data (u0, v0, w0) ∈ X0. If µ > 0 and there exists (t0, x0) ∈ R2 with either u(t0, x0) = 0 orv(t0, x0) = 0, then, by the strong maximum principle,
u(t, x) = 0 and v(t, x) = 0 for all (t, x)∈R2.
On the other hand, if µ = 0 and there exists (t0, x0) ∈ R2 with u(t0, x0) = 0 (resp.
v(t0, x0) = 0), then
u(t, x) = 0 (resp. v(t, x) = 0) for all (t, x)∈R2.
In order to state our key lemma, it is also convenient to introduce some new function setω0(c2, c1), which roughly stands for the closure (with respect to the locally uniform topology) of the orbits of the solution in moving frames with speeds in the interval [c2, c1]. More precisely:
Definition 3.1. For any (u0, v0, w0)∈X0, and any 0≤c2 < c1, we define the set ω0(c2, c1) :=closure{(u, v, w)(t,·+ct)| t≥0, c∈[c2, c1]} ⊂X0,
where (u, v, w) denotes the solution of (1.1)-(1.3) with the initial data (u0, v0, w0).
Herein the closure is understood with respect to the locally uniform topology inR. We point out that the fact that ω0(c2, c1) is a subset of X0 is a straightforward consequence of the above discussion together with standard parabolic estimates. In the sequel we have to keep in mind that this set depends on the initial data and, for notational convenience we omit to explicitly write down the dependence with respect to the initial data.
We now turn to some important lemma that shall be used several times throughout this manuscript.
Lemma 3.2. Assume that (u0, v0, w0) ∈ X0. Let 0 ≤ c2 < c1 and ζ, ξ ≥ 0 be given such that ζ+ξ > 0. Assume that for any c ∈ [c2, c1) there exists ε(c) >0 such that for any(˜u0,˜v0,w˜0)∈ω0(c2, c1)with ζu˜0+ξv˜0 6≡0, the corresponding solution (˜u,v,˜ w)˜ satisfies
(3.2) lim sup
t→+∞ (ζu(t, ct) +˜ ξ˜v(t, ct))≥ε(c).
Then for any c∈[c2, c1) the solution (u, v, w) satisfies:
(3.3) lim inf
t→+∞ inf
c2t≤x≤ct(ζu(t, x) +ξv(t, x))>0, as well as
(3.4) lim sup
t→+∞ sup
c2t≤x≤ct
w(t, x)<1.
Proof. The proof of this lemma involves three main steps described below.
First step. In this first step we shall show that for any c ∈ [c2, c1) there exists ε1(c)>0 such that
(3.5) lim inf
t→+∞ (ζu(t, ct) +ξv(t, ct))≥ε1(c).
To prove the above lower estimate we argue by contradiction by fixingc∈[c2, c1) and by assuming (using (3.2) for (u, v, w)) that there exist a sequence tn → +∞ and a sequencesn> tn such that for all n large enough
[ζu+ξv](tn, cntn) = ε(c)2 ,
[ζu+ξv](t, cnt)≤ ε(c)2 , ∀t∈[tn, sn], [ζu+ξv](sn, cnsn)≤ n1.
Then, possibly up to a sub-sequence not relabelled, one may assume that (u, v, w)(t+tn, x+ctn)→(u∞, v∞, w∞)(t, x),
asn→+∞, where the limit functions (u∞, v∞, w∞) solve (1.1)-(1.3) for allt ∈Rand (u∞, v∞, w∞)(0,·)∈ω0(c2, c1).
We also have by construction that
ζu∞(0,0) +ξv∞(0,0) = ε(c) 2 ,
so that, by the strong maximum principle,ζu∞+ξv∞>0. Furthermore let us notice that sn−tn → +∞ as n →+∞. Indeed, if the sequence {sn−tn} has a converging sub-sequence toσ ∈R, then the limit functions satisfy
ζu∞(σ, cσ) +ξv∞(σ, cσ) = 0.
So that if for instanceζu∞(0,0)>0 – henceζ >0 – then the previous equality ensures that u∞(σ, cσ) = 0 and u∞(t, x) = 0 for any (t, x)∈R2, a contradiction.
As a consequence, since sn−tn→+∞ asn →+∞, one obtains that ζu∞(t, ct) +ξv∞(t, ct)≤ ε(c)
2 , ∀t≥0.
Recalling that (u∞, v∞, w∞)(0,·) ∈ ω0(c2, c1) and ζu∞(0,·) +ξv∞(0,·) 6≡ 0, this con- tradicts (3.2) and completes the proof of (3.5).
Second step. We now prove (3.3). To that aim we consider the shifted function (ˆu,v,ˆ w)(t, x) := (u, v, w)(t, xˆ +c2t). Once more, we proceed by contradiction. Let
˜
c∈(c2, c1) be given. Assume that there exist tn →+∞, cn ∈[0,c˜−c2) such that
n→lim+∞(ζu+ξv)(tn,(c2+cn)tn) = lim
n→+∞(ζuˆ+ξˆv)(tn, cntn) = 0.
Without loss of generality, up to a sub-sequence, we assume that cn →c∈[0,c˜−c2].
Choose c′ such that c < c′ < c1−c2 and define the sequence t′n := cntn
c′ ∈[0, tn), ∀n≥0.
Consider first the case when the sequence {cntn} is bounded which may happen if c= 0. Then up to extraction of a sub-sequence, one has as n→+∞ that
cntn →x∞∈R, and, due to the strong maximum principle,
n→lim+∞(ζuˆ+ξˆv)(tn+t, cntn+x) = 0 locally uniformly for (t, x)∈R2.
This implies in particular that (ζuˆ+ξˆv)(tn,0) = (ζu+ξv)(tn, c2tn)→0 asn →+∞, which contradicts (3.5) withc=c2. As a consequencec > 0, the sequence{cntn} has no bounded sub-sequence and we can assume below thatt′n→+∞as n→+∞.
Now due to (3.5) we have for all large n that
(ζuˆ+ξv)(tˆ ′n, cntn) = (ζuˆ+ξˆv)(t′n, c′t′n) = (ζu+ξv)(t′n,(c2+c′)t′n)> 3
4ε1(c2+c′).
Next, we introduce a third time sequence{t′′n} by t′′n := inf
{
t≤tn| ∀s∈(t, tn), (ζuˆ+ξˆv)(s, cntn)≤ min{ε(c2), ε1(c2+c′)} 2
}
∈(t′n, tn).
Since (ζuˆ+ξˆv)(tn, cntn)→0 as n→+∞, we get
(ζuˆ+ξv)(tˆ ′′n, cntn) = min{ε(c2), ε1(c2+c′)}
2 ,
and, as before, by a limiting argument and a strong maximum principle, that tn−t′′n →+∞
asn →+∞.
Then, by parabolic estimates and up to extraction of a sub-sequence, we find that (ˆu,v,ˆ w)(tˆ +t′′n, x+cntn−c2t) converges to a solution (u∞, v∞, w∞) such that
(ζu∞+ξv∞)(0,0)>0,
(ζu∞+ξv∞)(t, c2t)≤ min{ε(c2), ε1(c2+c′)}
2 , ∀t≥0.
(3.6)
Note also that (ˆu,v,ˆ w)(tˆ +t′′n, x+cntn−c2t) = (u, v, w)(t+t′′n, x+cntn+c2t′′n). Hence, since 0≤cntn=c′t′n≤c′t′′n ≤(c1−c2)t′′n, one gets
xn:=cntn+c2t′′n ∈[c2t′′n, c1t′′n],
so that (u∞, v∞, w∞)(0,·)∈ω0(c2, c1) and (3.6) contradicts (3.2). This completes the proof of (3.3).
Third step. Finally we deal with (3.4) and the w component. We again proceed by contradiction. Let c ∈ [c2, c1) be given and assume that w(tn, xn) → 1 for some sequences tn → +∞ and c2tn ≤xn ≤ctn. Then, up to extraction of a sub-sequence, (u, v, w)(t +tn, x+xn) converges to an entire in time (i.e., for all t ∈ R) solution (u∞, v∞, w∞) of (1.1)-(1.3), which also satisfies
w∞(0,0) = 1.
It follows from the strong maximum principle thatw∞ ≡1. However, according to the above we haveζu∞+ξv∞>0, hence eitheru∞>0 orv∞>0. This is a contradiction
to (1.3) and the lemma is proved.
The above lemma can be strengthened as follows in the coupling case µ >0.
Corollary 3.3. Assume thatµ > 0and(u0, v0, w0)∈X0. Letc1 >0be given. Assume also that for anyc∈[0, c1)there existsε(c)>0such that for any(˜u0,v˜0,w˜0)∈ω0(0, c1) with u˜0+ ˜v0 6≡0, the corresponding solution (˜u,v,˜ w)˜ satisfies
lim sup
t→+∞ (˜u(t, ct) + ˜v(t, ct))≥ε(c).
Then for any c∈[0, c1) the solution (u, v, w) satisfies
(3.7) lim inf
t→+∞ inf
0≤x≤ctmin{u(t, x), v(t, x),1−w(t, x)}>0.
The proof of this result follows from a straightforward limiting argument together with the strong maximum principle. Indeed, for any c ∈ [0, c1), from Lemma 3.2 it follows that
(3.8) lim inf
t→+∞ inf
0≤x≤ctmin{u(t, x) +v(t, x),1−w(t, x)}>0.
Next, as in the third step of the proof of Lemma 3.2, if there exist sequencestn →+∞ and 0 ≤ xn ≤ ctn such that u(tn, xn) →0 as n → +∞, then (u, v, w)(t+tn, x+xn) converges to an entire in time solution (0, v∞, w∞). From (3.8) we must have v∞>0, which contradicts (1.1). It follows that
lim inf
t→+∞ inf
0≤x≤ctu(t, x)>0.
Proceeding similarly for the other predator, one reaches the conclusion (3.7).
Remark 3.1. The same results as in Lemma 3.2 and Corollary 3.3 also hold for negative speeds with straightforward adaptations.
3.3. Some eigenvalue problems. We conclude this section by a computational lemma that shall be used in the sequel in particular to check the assumptions of Lemma 3.2 and Corollary 3.3.
Lemma 3.4. Let d > 0, c ∈ R, a ∈ R and R > 0 be given. Then the principal eigenvalue λR of the following Dirichlet elliptic problem
{−dφ′′(x)−cφ′(x) +aφ(x) =λRφ(x), for x∈(−R, R), φ(±R) = 0 and φ > 0 on (−R, R),
is given by
λR=a+ c2
4d+ dπ2 4R2. This is classical and we omit the proof.
In the ‘two mutants’ case, we need an analogous result for a (cooperative) system as follows.
Proposition 3.5. Letµ >0, c∈R, δ ≥0andR >0be given. Consider the following principal eigenvalue problem:
−d1φxx−cφx−r1φ(a−1−2δ)−µ(ψ−φ) = ΛRφ, for x∈(−R, R),
−d2ψxx−cψx−r2ψ(a−1−2rr1
2δ)−µ(φ−ψ) = ΛRψ, for x∈(−R, R), φ(±R) =ψ(±R) = 0 and φ, ψ >0 on (−R, R).
Then there exists a uniqueΛR(c, δ)such that this eigenvalue problem admits an eigen- function pair (φ, ψ) with φ, ψ > 0 in (−R, R). It satisfies ΛR(c, δ) = ΛR(c,0) + 2r1δ andΛR(−c,0) = ΛR(c,0). Moreover, for each|c|< c∗µ, there exist δ0 >0and R0 such that
ΛR(c, δ)<0, ∀δ∈[0, δ0], ∀R ≥R0.
Since this result was proved in [12], we only give a brief sketch of the argument.
Proof. First, by the classical Krein-Rutman theory, there exists a unique ΛR(c, δ) such that this eigenvalue problem admits a positive eigenfunction pair. The symmetry with respect to c follows from the uniqueness of this eigenvalue and by changing x to −x into the system.
We consider now 0 ≤ c < c∗µ. As R → +∞, ΛR(c, δ) converges to a generalized principal eigenvalue of the operator
A[φ, ψ] :=
( −d1φxx−cφx−r1φ(a−1−2δ)−µ(ψ−φ)
−d2ψxx−cψx−r2ψ(a−1−2rr1
2δ)−µ(φ−ψ) )
;
see [12, Theorem 4.2]. More precisely, this generalized principal eigenvalue is defined as
sup{λ ∈R| ∃(φ, ψ)∈C2(R,R∗+×R∗+), A[φ, ψ]≥λ[φ, ψ]},
where the inequality is to be understood componentwise. It also turns out (see [12, Lemma 6.4]) that this generalized principal eigenvalue coincides with the maximum of the function
γ ∈[0,+∞)7→ −Λ[µ, γ] + 2r1δ+cγ,
where Λ[µ, γ] is the Perron-Frobenius eigenvalue of the matrix defined by (2.3). In other words, one has uniformly with respect to δ that
R→lim+∞ΛR(c, δ) = max
γ≥0 {−Λ[µ, γ] +cγ}+ 2r1δ.
Recalling that Λ[µ,0]>0,
c∗µ := min
γ>0
Λ[µ, γ]
γ >0,
and 0≤c < c∗µ, one gets the existence of R0 >0 large enough andδ0 >0 small enough such that for allR ≥R0 and δ ∈[0, δ0] one has
ΛR(c, δ)<0.
The proposition is proved.
We point out that it follows from this proposition that solutions of the sub-system { ut =d1uxx+r1u(a−1−u−kv) +µ(v −u),
vt=d2vxx+r2v(a−1−hu−v) +µ(u−v),
which arises from taking w ≡ 1 in (1.1)-(1.2), spread with speed c∗µ when 0 < µ ≤ (a−1) min{r1, r2}/2. We refer to Girardin [12] for a proof of this result, in a more general framework including an arbitrary number of mutants. This relies on the fact that this sub-system roughly has a cooperative structure around the trivial steady state (0,0), i.e., the linearized system around (0,0) is cooperative, since competitive terms are nonlinear. As a matter of fact, we believe that our analysis for the predator- prey system could also work when we increase the number of mutants. As we shall see later, our proof relies on a similar construction of subsolutions as in [12], hence on the above proposition.
4. Uniformly positive entire solutions
In this section, we state some Liouville type results on the stationary solutions of both the full ‘two competitors’ system and its sub-systems with only one predator.
This relies on a Lyapunov approach and it shall allow us to describe the shape of the solutions behind the propagation fronts. Throughout this section we shall assume that
µ= 0.
4.1. Two-dimensional sub-systems. We start with an important lemma on bounded and uniformly positive entire solutions of the sub-system
{
ut−d1uxx =uF(u,0, w),
wt−d3wxx =wH(u,0, w), t∈R, x∈R. Our result reads as follows.
Lemma 4.1. Let (u, w) = (u, w)(t, x) be an entire solution of the above system such that there exist constants M > m >0 with
m ≤u(t, x)≤M, m≤w(t, x)≤M, ∀(t, x)∈R2. Then (u, w)≡(˜p,q), where the stationary state˜ (˜p,q)˜ is defined in (2.1).
Before we prove this result, we point out that the roles of u can easily be inverted.
In particular, considering {
vt−d2vxx =vG(0, v, w),
wt−d3wxx =wH(0, v, w), t∈R, x∈R, we also have:
Lemma 4.2. Let (v, w) = (v, w)(t, x) be an entire solution of the above system such that there exist constants M > m >0 with
m≤v(t, x)≤M, m≤w(t, x)≤M, ∀(t, x)∈R2. Then (v, w)≡(˜p,q), where the stationary state˜ (˜p,q)˜ is defined in (2.1).
Proof. Since both results are actually the same up to renaming parameters, we only prove Lemma 4.1. In order to prove this lemma we make use of a Lyapunov like argument, inspired from [4, 7].
Define the function g(x) = x−ln(x)−1. Recalling the definition of (˜p,q) in (2.1)˜ one has
F(u,0, w) =r1[˜p−u+a(w−q)], H(u,˜ 0, w) =r2[b(˜p−u) + (˜q−w)].
Consider also the functionV =V(u, w) given by V(u, w) = br2pg˜
(u
˜ p
)
+ar1qg˜ (w
˜ q
)
:=V1(u) +V2(w).
Note that the Lie Derivative of V along X = (uF(u,0, w), wH(u,0, w)), denoted by LXV, is given by
LXV(u, w) = Vu(u, w)uF(u,0, w) +Vw(u, w)wH(u,0, w)
=br1r2(u−p) [˜˜ p−u+a(w−q)] +˜ ar1r2(w−q)[b(˜˜ p−u) + (˜q−w)]
=−br1r2(u−p)˜2−ar1r2(w−q)˜2.
Now observe that there exists some constantα >0 such that
LXV(u, w)≤ −αV(u, w), ∀(u, w)∈[m, M]×[m, M].
