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Fronts from two-dimensional dispersal kernels: Beyond the nonoverlapping-generations model

Daniel R. Amor and Joaquim Fort

  • Departament de Física, Universitat de Girona, Girona, 17071 Catalonia, Spain

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Phys. Rev. E 80, 051918 – Published 23 November, 2009

DOI: https://doi.org/10.1103/PhysRevE.80.051918

Abstract

Most integrodifference models of biological invasions are based on the nonoverlapping-generations approximation. However, the effect of multiple reproduction events (overlapping generations) on the front speed can be very important (especially for species with a long life spam). Only in one-dimensional space has this approximation been relaxed previously, although almost all biological invasions take place in two dimensions. Here we present a model that takes into account the overlapping generations effect (or, more generally, the stage structure of the population), and we analyze the main differences with the corresponding nonoverlapping-generations results.

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  24. According to field observations in sites close to those where the dispersal kernel was measured, we estimate the net reproductive rate R0 of the yellow poplar (Liriodendron Tulipifera) to be in the range of 2–50 seeds/(tree yr.). The age at first reproduction (generation time in the nonstructured model) of the same species is T=20 yr. As shown in [10], the long-distance dispersal component of the kernel has a much more important effect on the front speed than the short-distance component (even if long-distance dispersal happens seldom). Therefore, we use for the distance between cells that from the long-distance component, Δ0=6000 m, both in the CSRW and in the DSRW models.
  25. All of the results in this paper have been calculated with square matrices A⃗⃗ and Φ⃗⃗ of order 130. However, we have observed that in many cases we can obtain accurate results using matrices of lower order. That can be very useful in order to reduce computation times. For instance, omitting all of the elements aij with i>65 and/or j>65, we obtain a demographic matrix of order 65. If we do the same for the dispersal kernel matrix and we solve the equations of our overlapping-generations model, we obtain the same invasion speeds (differences are under the 1%) for values of R0>5. The reason is that very old individuals do not appear near the leading edge of the invasion front, so for high values of the fecundity R0 the front dynamics is driven by younger individuals and the contribution of older ones is not relevant. However we do not give a general description of such an approximation because it depends on the dispersal kernel and demographic characteristics of each species.
  26. Calculation of each overlapping-DSRW speed in Fig. 2 takes about 50 to 80 minutes. We used an Intel Core 2 CPU, T7400 (2,16GHz and 2GB RAM). We remark that calculation times become much longer as the order of the matrix H⃗⃗ increases. In contrast, each overlapping-model simulation takes about 10 minutes using the same computer. For nonoverlapping models, calculation times are about ten times shorter than for overlapping models.

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Phys. Rev. E 80, 051918– Published 23 November, 2009
  • Received 30 July 2009

© 2009 American Physical Society

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