Bulletin of Mathematical Biology

, Volume 74, Issue 2, pp 453–473

# Success rate of a biological invasion in terms of the spatial distribution of the founding population

Original Article

DOI: 10.1007/s11538-011-9694-9

Garnier, J., Roques, L. & Hamel, F. Bull Math Biol (2012) 74: 453. doi:10.1007/s11538-011-9694-9

## Abstract

We analyze the role of the spatial distribution of the initial condition 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 α. Analytical as well as numerical results indicate that the critical size L(α) of the population, where the invasion is successful if and only if L>L(α), is a continuous function of α and tends to increase with α, at least when α 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 u0, and we provide a new and rigorous definition of the rate of fragmentation of u0. We then conduct a statistical analysis of the probability of successful invasion depending on the size of the support of u0 and the fragmentation rate of u0. 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 non-linear fashion with the fragmentation rate. This effect of fragmentation is enhanced as the strength of the Allee effect is increased.

### Keywords

Reaction–diffusion Allee effect Biological invasions Fragmentation Initial condition

© Society for Mathematical Biology 2011

## Authors and Affiliations

• Jimmy Garnier
• 1
• 2
• Lionel Roques
• 1
• François Hamel
• 2
• 3
1. 1.UR 546 Biostatistique et Processus SpatiauxINRAAvignonFrance
2. 2.LATP, Faculté des Sciences et TechniquesAix-Marseille UniversitéMarseille Cedex 20France
3. 3.Institut Universitaire de FranceParisFrance