[Paper Review] The diffusive competition problem with a free boundary in heterogeneous time-periodic environment

Qiaoling Chen, Fengquan Li+1 more|arXiv (Cornell University)|Apr 21, 2015

Mathematical and Theoretical Epidemiology and Ecology Models21 references3 citations

TL;DR

This paper studies a diffusive competition model with a free boundary in a time-periodic, heterogeneous environment, where an invasive species (u) spreads from an initial region while competing with a native species (v). It establishes a spreading-vanishing dichotomy and provides sufficient conditions for spreading or extinction, along with rough estimates of spreading speed using periodic solutions of auxiliary problems.

ABSTRACT

In this paper, we consider the diffusive competition problem with a free boundary and sign-changing intrinsic growth rate in heterogeneous time-periodic environment, consisting of an invasive species with density $u$ and a native species with density $v$. We assume that $v$ undergoes diffusion and growth in $R^{N}$ , and $u$ exists initially in a ball $B_{h_0}(0)$, but invades into the environment with spreading front $\{r = h(t)\}$. The effect of the dispersal rate $d_1$, the initial occupying habitat $h_0$, the initial density $u_0$ of invasive species $u$, and the parameter $μ$ (see (1.3)) on the dynamics of this free boundary problem are studied. A spreading-vanishing dichotomy is obtained and some sufficient conditions for the invasive species spreading and vanishing are provided. Moreover, when spreading of $u$ happens, some rough estimates of the spreading speed are also given.

Motivation & Objective

• To understand the long-term dynamics of an invasive species spreading into a habitat occupied by a native competitor in a time-periodic, heterogeneous environment.
• To examine how dispersal rate $ d_1 $, initial habitat size $ h_0 $, initial density $ u_0 $, and competition parameter $ \mu $ affect the outcome of invasion.
• To establish a spreading-vanishing dichotomy for the free boundary problem with sign-changing intrinsic growth rates.
• To derive sufficient conditions for the invasive species to successfully spread or vanish.
• To estimate the asymptotic spreading speed of the invasive species when spreading occurs.

Proposed method

• Formulates a reaction-diffusion system with a free boundary $ h(t) $, where the spreading front evolves via a Stefan-type condition $ h'(t) = -\mu u_r(t, h(t)) $.
• Imposes time-periodic and Hölder-continuous coefficients for growth, self-limitation, and competition terms, satisfying upper and lower bounds.
• Uses comparison principles and monotonicity arguments to compare the solution with auxiliary problems having constant or periodic coefficients.
• Analyzes the existence and uniqueness of positive T-periodic solutions to a related logistic equation with time-periodic coefficients.
• Introduces a function $ K_0(\mu, a, b) $ that characterizes the spreading speed in the absence of competition, derived from the periodic solution of a scalar ODE.
• Applies the function $ K_0 $ to bound the spreading speed of $ h(t) $ from above and below using extremal coefficient functions.

Experimental results

Research questions

• RQ1 Under what conditions does the invasive species $ u $ spread or vanish in the heterogeneous time-periodic environment?
• RQ2 How do the dispersal rate $ d_1 $, initial habitat size $ h_0 $, initial density $ u_0 $, and parameter $ \mu $ influence the spreading or vanishing outcome?
• RQ3 What is the asymptotic spreading speed of the invasive species when it successfully spreads?
• RQ4 How does the presence of the native species $ v $, which evolves on the entire space, affect the spreading dynamics of $ u $?
• RQ5 Can the spreading speed of $ u $ be estimated using periodic solutions of auxiliary problems with extremal coefficients?

Key findings

• A spreading-vanishing dichotomy holds: either the invasive species $ u $ spreads indefinitely (i.e., $ h(t) \to \infty $) or it vanishes (i.e., $ h(t) \to h_\infty < \infty $).
• Sufficient conditions for spreading are derived based on the initial habitat size $ h_0 $, initial density $ u_0 $, and the parameters $ d_1 $ and $ \mu $, particularly when the effective growth rate is sufficiently positive.
• When spreading occurs, the spreading speed satisfies the inequality $ \frac{1}{T}\int_0^T K_0(\mu, m_{1,*} - c_1^* V^*, b_1^*) \, dt \leq \liminf_{t\to\infty} \frac{h(t)}{t} \leq \limsup_{t\to\infty} \frac{h(t)}{t} \leq \frac{1}{T}\int_0^T K_0(\mu, m_1^*, b_{1,*}) \, dt $.
• The function $ K_0(\mu, a, b) $ is well-defined and continuous in its arguments, and satisfies $ 0 < \frac{1}{T}\int_0^T K_0(\mu, a, b) \, dt < 2\sqrt{\frac{d}{T}\int_0^T a(t) \, dt} $, ensuring boundedness of the spreading speed.
• The periodic solution $ U^K $ of the auxiliary problem is positive, strictly increasing in $ r $, and converges uniformly to a positive T-periodic solution $ V(t) $ as $ r \to \infty $.
• If $ K_1 \leq K $ and $ K_1 \not\equiv K $, then $ U^{K_1}(t,r) > U^K(t,r) $ and $ U^{K_1}_r(t,0) > U^K_r(t,0) $, showing monotonicity with respect to the boundary flux.