A reaction-diffusion-advection competition model with a free boundary (original) (raw)
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Discrete and Continuous Dynamical Systems - Series B, 2014
In this paper we consider the diffusive competition model consisting of an invasive species with density u and a native species with density v, in a radially symmetric setting with free boundary. We assume that v undergoes diffusion and growth in R N , and u exists initially in a ball {r < h(0)}, but invades into the environment with spreading front {r = h(t)}, with h(t) evolving according to the free boundary condition h (t) = −µur(t, h(t)), where µ > 0 is a given constant and u(t, h(t)) = 0. Thus the population range of u is the expanding ball {r < h(t)}, while that for v is R N . In the case that u is a superior competitor (determined by the reaction terms), we show that a spreading-vanishing dichotomy holds, namely, as t → ∞, either h(t) → ∞ and (u, v) → (u * , 0), or limt→∞ h(t) < ∞ and (u, v) → (0, v * ), where (u * , 0) and (0, v * ) are the semitrivial steady-states of the system. Moreover, when spreading of u happens, some rough estimates of the spreading speed are also given. When u is an inferior competitor, we show that (u, v) → (0, v * ) as t → ∞, so the invasive species u always vanishes in the long run.
On a competitive system with ideal free dispersal
Journal of Differential Equations, 2018
In this article we study the long term behavior of the competitive system ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂u ∂t = ∇ • α(x)∇ u m + u(m(x) − u − bv) in , t > 0, ∂u ∂t = ∇ • [β(x)∇v] + v(m(x) − cu − v) in , t > 0, ∇ u m •n = ∇v •n = 0 on ∂ , t > 0, which supports for the first species an ideal free distribution, that is a positive steady state which matches the per-capita growth rate. Previous results have stated that when b = c = 1 the ideal free distribution is an evolutionarily stable and neighborhood invader strategy, that is the species with density v always goes extinct. Thus, of particular interest will be to study the interplay between the inter-specific competition coefficients b, c and the diffusion coefficients α(x) and β(x) on the critical values for stability of semi-trivial steady states, and the structure of bifurcation branches of positive equilibria arising from these equilibria. We will also show that under certain regimes the system sustains multiple positive steady states.
A diffusive competition model with a protection zone
Journal of Differential Equations, 2008
This paper is concerned with a two species diffusive competition model with a protection zone for the weak competitor. Our mathematical results imply that when the protection zone is above a certain critical patch size determined by the birth rate of the weak competitor, the weak species almost always survives, but it cannot survive when the protection zone is below the critical size and its competitor is strong enough. While this is the main feature of the model, the actual dynamical behavior of the reaction-diffusion system is more complicated. The key to reveal the main feature of the system lies in a detailed analysis of the attracting regions of its steady-state solutions. Our mathematical analysis shows that, compared with the predator-prey model discussed in [Yihong Du, Junping Shi, A diffusive predator-prey model with a protect zone, J. Differential Equations 226 (2006) 63-91], the protection zone has some essentially different effects on the fine dynamics of the competition model.
Singular limit of a competition–diffusion system with large interspecific interaction
Journal of Mathematical Analysis and Applications, 2012
We consider a competition-diffusion system for two competing species; the density of the first species satisfies a parabolic equation together with a inhomogeneous Dirichlet boundary condition whereas the second one either satisfies a parabolic equation with a homogeneous Neumann boundary condition, or an ordinary differential equation. Under the situation where the two species spatially segregate as the interspecific competition rate becomes large, we show that the resulting limit problem turns out to be a free boundary problem. We focus on the singular limit of the interspecific reaction term, which involves a measure located on the free boundary.
Journal of Mathematical Biology, 2018
Reaction-diffusion systems with a Lotka-Volterra-type reaction term, also known as competition-diffusion systems, have been used to investigate the dynamics of the competition among m ecological species for a limited resource necessary to their survival and growth. Notwithstanding their rather simple mathematical structure, such systems may display quite interesting behaviours. In particular, while for m = 2 no coexistence of the two species is usually possible, if m ≥ 3 we may observe coexistence of all or a subset of the species, sensitively depending on the parameter values. Such coexistence can take the form of very complex spatio-temporal patterns and oscillations. Unfortunately, at the moment there are no known tools for a complete analytical study of such systems for m ≥ 3. This means that establishing general criteria for the occurrence of coexistence appears to be very hard. In this paper we will instead give some criteria for the non-coexistence of species, motivated by the ecological problem of the invasion of an ecosystem by an exotic species. We will show that when the environment is very favourable to the invading species the invasion will always be successful and the native species will be driven to extinction. On the other hand, if the environment is not favourable enough, the invasion will always fail. Keywords competition-diffusion system • ecological invasion • competitive exclusion • large-time behaviour • singular limit • comparison principle Mathematics Subject Classification 35Q92 • 92D25 • 35K57 • 35B25 • 35B40 • 35B51
Free boundary problem for predator-prey model
E3S Web of Conferences
In this article, we study the behavior of two species evolving in a domain with a free boundary. This system mimics the spread of invasive or new predator species, in which free boundaries represent the expanding fronts of predator species and are described by the Stefan condition. A priori estimates for the required functions are established. These estimates are used to prove the existence and uniqueness of the solution.
A Reaction-Advection-Diffusion Model in Spatial Ecology: Theoretical and Computational Analysis
Khulna University Studies
In a confined heterogeneous habitat with two species interacting for common resources, the research analyzes a reaction-advection-diffusion type dispersal model with homogeneous Neumann boundary conditions for generalized growth functions. Both species follow the same symmetric growths law, but their dispersal strategies and advection rates are different. The following pattern is used to consider the competition strategy: in a bounded heterogeneous habitat, the first population disperses according to its resource functions, whereas the second population disperses according to its carrying capacity. We investigate the model in two scenarios: when carrying capacity and resource functions are non-proportional, competitive exclusion occurs, and one species drives theother to extinction in the long run for various similar and unequal carrying capacities of competing species. However, coexistence is achievable for different resource distribution consumption if the resource distribution an...
Limiting behaviour for a prey-predator model with diffusion and crowding effects
Journal of Mathematical Biology, 1978
where a~, ~2, a, b, c, p, q, r are positive constants related to diffusion, growth, interaction and death rates of the populations ux, u2. The unknown population functions Ul, u2 are defined for time t >/0 and space x = (xx ..... x~) e ~ (f2 is a given bounded open connected subset of R n, n >/ 1). When u~ are functions of t alone, (1.1) reduces to the generalized Volterra-Lotka model for prey-predator interactions, with b, q expressing the crowding effect on the growth of prey ul and predator u2 respectively. The parameters a, r, c,p represent growth, death and interaction rates. (See e.g. Poole [5] or Rescigno and Richardson [7]). With the inclusion of diffusion terms a~Aul, a2Au2, (A _= ~=1 ~2/~x2), Williams and Chow [8] studied Eq. (1.1) with no crowding effect (i.e. b = q = 0). Since the effects of crowding tend to occur in realistic biological systems, this article studies Eq. (1.1) with b > 0, q > 0, extending the results in [8]. We will analyze Eq. (1.1) with the condition that no diffusion occurs across the boundary of the space domain ~ (see (2.2)). Firstly, Theorem 1 in Section 2 quotes a theorem due to Williams and Chow [8], asserting that u~(x, t) will remain positive for t > 0 under appropriate initial-boundary conditions. Note that this preliminary theorem is true, even if b and q are positive. We will finally show that u~(x, t), i = l, 2, will tend to an equilibrium solution which is homogeneous in space, as t-> +~. An appropriate energy function will be constructed for proving this result. Further, it will be seen that the integral of the energy function over space will be a monotonically non-increasing function of time t. These observations should have significant implications concerning experimental designs for biologists and ecologists. Furthermore, they can contribute to saving tremendous amount of numerical computational efforts. One should also note that the energy function
Instability of non-constant equilibrium solutions of a system of competition-diffusion equations
Journal of Mathematical Biology, 1981
The system of interaction-diffusion equations describing competition between two species is investigated. By using a version of the Perron-Frobenius theorem of positive matrices generalized to function spaces, it is proved that any non-constant equilibrium solution of the system is unstable both under Neumann boundary conditions (for the rectangular parallelepiped domain) and under periodic conditions. It is conjectured that this result extends to convex domains, and that the simple interaction-diffusion model cannot explain spatially segregated distributions of two competing species in such domains.