The minimal habitat size for spreading in a weak competition system with two free boundaries. (English) Zbl 1319.35081

In this paper, the dynamics of a Lotka-Volterra type weak competition system \[ \begin{aligned} u_t &= d_1u_{xx}+r_1u(1-u-kv),\\ v_t &= d_2v_{xx} +r_2v(1-v-hu) \end{aligned} \] is investigated, where \(u_x(0, t) = v_x(0, t) = 0\), \(u \equiv 0\) for \(x\geq s(t)\) and \(v\equiv 0\) for \(x \geq \sigma(t)\), and \(s(t)\) and \(\sigma(t)\) satisfy certain equations. While the coefficients \(k, h\) reflect the competition between the two species \(u(x, t)\) and \(v(x, t)\), the restriction \(0 < k\), \(h < 1\) reflects the case of weak competition. This paper continues the author’s previous work with others (for \(0 < k < 1 < h\)) in the case of weak competition. With the classification of the dynamics into four cases, it establishes a spreading-vanishing quartering. Some criteria are provided for each case to happen, estimates for the spreading speed and the long-term behaviour of solutions are established, the notion of minimal habitat size is introduced for successful spreading, and a sharp criterion is also established for species spreading and vanishing.


35K51 Initial-boundary value problems for second-order parabolic systems
35R35 Free boundary problems for PDEs
92B05 General biology and biomathematics
Full Text: DOI


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