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Spreading with two speeds and mass segregation in a diffusive competition system with free boundaries. (English) Zbl 1396.35028

The authors investigate the spreading behaviors of two invasive species modeled by a Lotka-Volterra diffusive competition system with two free boundaries: \[ \begin{aligned} {u_t} & = d\Delta u + ru(1 - u - kv),\;r \in (0,{s_1}(t)),\;t > 0\\ {v_t} & = \Delta v + v(1 - v - hu),\;r \in (0,{s_2}(t)),\;t > 0\\ {u_r}(t,0) & = {v_r}(t,0) = 0,\;t > 0\\ {s_1}(0) & = s_1^0,\;{s_2}(0) = s_2^0,\;u(0,r) = {u^0}(r),\;v(0,r) = {v^0}(r),\;r \in [0,\infty ) \end{aligned} \] where \(d,h,k > 0\) are constants, \(u(t,r)\) and \(v(t,r)\) represent the population densities of the two competing species at spatial location \(\text{{r = |x|,}}\;\text{{x}} \in {^n},\;n \geq 2\), \(\Delta \varphi = {\varphi _{rr}} + \frac{{n - 1}}{r}{\varphi _r}\) is the usual Laplace operator acting on spherically symmetric functions. In the considerations is assumed that \(0 < k < 1< h\) (which is often referred to as a weak-strong competition case). In the investigated case under suitable assumptions for the considered model is established that the both species \(u\) and \(v\) survive in the competition in the system, can successfully spread into the available environment with different spreading speeds and their population masses tend to segregate, with the population mass of shifting to infinity as \(\text{{t}} \to \infty\).

MSC:

35K51 Initial-boundary value problems for second-order parabolic systems
35J87 Unilateral problems for nonlinear elliptic equations and variational inequalities with nonlinear elliptic operators
35Q92 PDEs in connection with biology, chemistry and other natural sciences
92B05 General biology and biomathematics
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