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Abstract
For a large class of models involving several species competing for a single resource in a homogeneous environment, it is known that the competitive exclusion principle (CEP) holds: only one species survives eventually. Various works indicate though that coexistence of many species is possible when the competition occurs in a heterogeneous environment. We propose here a spatially heterogeneous system modeling several species competing for a single resource, and migrating in the spatial domain. For this model, it is known, at least in particular cases, that if migrations are slow enough, then coexistence occurs. In this paper, we show at variance that if the spatial migrations are fast enough, then our system can be approximated by a spatially homogeneous system called aggregated model, which can be explicitly computed, and we show that if the CEP holds for the aggregated model, then it holds as well for the original, spatially heterogeneous model. In other words, we show the persistence of the CEP in the spatially heterogeneous situation when migrations are fast. As a consequence, for fast migrations only one species may survive, namely the best competitor in average. We last study which is the best competitor in average on some examples, and draw some ecological consequences.
Notes
1 The key to apply this method is as follows: (1) There is an control on the solutions uniformly in time, of the form
. (2) The system has a particular structure. For our system, the system is triangular since the
are coupled indirectly through
. (3) There is a uniform bound for a (well chosen) component of
. Here,
.
2 Theorem 2.5 holds true with an initial condition . However, since
belongs to
for any
, we reduce ourselves to the case of continuous initial data. This will simplify the statement of the main results. Finally, the solution is more regular since
for any
.
3 In the case of more general operators (see Remark 2.1), one has for some positive function
and
. For the sake of simplicity, we reduce ourselves to the case of operator
, for which
and
do not depends on
.
4 That is, is not an eigenvalue of
.
5 A hyperbolic solution is say to be (linearly) asymptotically stable (resp. unstable) if the real part of all the eigenvalues of
is negative (resp. if the real part of at least one eigenvalue is positive). In the sequel, we shall always refer to linear stability or instability.
6 The Proposition 2.9 holds true under more general assumptions, see the monograph of Smith and Waltmann [Citation3]. Indeed, a well-known conjecture asserts that the CEP holds true under the simpler hypothesis of monotonicity of the functions . This result is proved for equal mortalities in Armstrong and McGehee [Citation32]. In the case of different mortalities, this result is proved using Lyapunov functionals when the functions
verify some additional assumptions. We refer to Hsu [Citation33], Wolkowicz and Lu [Citation4], Wolkowicz and Xia [Citation5] and Li [Citation1] for historical advances on this topic. See also Sari and Mazenc [Citation2] for recent results on this subject.
7 Indeed, admits an asymptotic expansion of the form
which is explicitly computable provided the functions
and
have
smoothness. The approximation
leads to writing reduced systems of order
(see [Citation22]). This paper focuses only on the case
.
8 Indeed, this is a general fact for center manifold as pointed out by Carr [Citation17].
9 Let us remark at this step that one gets so that for any initial data
, one gets
for small enough
depending on
. It follows directly that the global asymptotic behavior holds true when
is far enough from the boundary. One can also reformulate this by saying that for any compact subset
of
, there exists
such that for any
, the global asymptotic behavior holds. The only problem occurs when
, which can happen in
.
10 Numerical evidence shows that a weak local competitor cannot survive to the competition for small diffusion rates. As it is proved in [Citation14], a rigourous study of stationary solutions for small diffusion supports these evidences.
11 As it is shown in [Citation34], stationary coexistence of species in
sites is generically impossible. Thus, three species cannot coexist in less than three patches.