Success rate of a biological invasion in terms of the spatial distribution of the founding population. (English) Zbl 1238.92046
Summary: We analyze the role of the spatial distribution of the initial conditions in reaction-diffusion models of biological invasion. Our study shows that, in the presence of an Allee effect, the precise shape of the initial (or founding) population is of critical importance for successful invasion. Results are provided for one-dimensional and two-dimensional models. In the one-dimensional case, we consider initial conditions supported by two disjoint intervals of length \(L/2\) and separated by a distance \(\alpha\). Analytical as well as numerical results indicate that the critical size \(L^{\ast}(\alpha)\) of the population, where the invasion is successful if and only if \(L > L^{\ast}(\alpha)\), is a continuous function of \(\alpha\) and tends to increase with \(\alpha\), at least when \(\alpha\) is not too small. This result emphasizes the detrimental effect of fragmentation.
In the two-dimensional case, we consider more general, stochastically generated initial conditions \(u_{0}\), and we provide a new and rigorous definition of the rate of fragmentation of \(u_{0}\). We then conduct a statistical analysis of the probability of successful invasion depending on the size of the support of \(u_{0}\) and the fragmentation rate of \(u_{0}\). Our results show that the outcome of an invasion is almost completely determined by these two parameters. Moreover, we observe that the minimum abundance required for successful invasion tends to increase in a nonlinear fashion with the fragmentation rate. This effect of fragmentation is enhanced as the strength of the Allee effect is increased.
In the two-dimensional case, we consider more general, stochastically generated initial conditions \(u_{0}\), and we provide a new and rigorous definition of the rate of fragmentation of \(u_{0}\). We then conduct a statistical analysis of the probability of successful invasion depending on the size of the support of \(u_{0}\) and the fragmentation rate of \(u_{0}\). Our results show that the outcome of an invasion is almost completely determined by these two parameters. Moreover, we observe that the minimum abundance required for successful invasion tends to increase in a nonlinear fashion with the fragmentation rate. This effect of fragmentation is enhanced as the strength of the Allee effect is increased.
MSC:
| 92D40 | Ecology |
| 92D15 | Problems related to evolution |
| 35K57 | Reaction-diffusion equations |
| 65C20 | Probabilistic models, generic numerical methods in probability and statistics |
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
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