Abstract
In order to look into the stability consequences of a particular migration process in which individuals choose to settle, we formulated a continuous-time multi-species multi-patch model in which individuals migrate by one or two instantaneous jumps while making the second jump with a certain probability that possibly depends on the conditions at the end point of the first jump. It turned out that a second jump has some quantitative effects on diffusive instability even when it occurs in the absence of density-dependent mechanisms. When a second jump happens as a natural interspecific response of individuals, and such a response is sufficiently strong, it has crucial effects on diffusive instability: it leads to diffusive instability in the case of competitive interactions, whereas it annihilates diffusive instability in the case of prey-predator interactions. We demonstrated these results in the context of two specific examples.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aronson, D. G. (1985). The role of diffusion in mathematical population biology: Skellam revisited, In Mathematics in Biology and Medicine, V. Capasso, (Ed.), Lecture Notes in Biomathematics 57, pp 2–6.
Diekmann, O. (1978). Thresholds and traveling waves for the geographical spread of infection. J. Math. Biol. 6, 109–130.
Diekmann, O. (1979). Run for your life: a note on the asymptotic speed of propagation of an epidemic. J. Differ. Equ. 33, 58–73.
Holt, R. D. (1984). Spatial heterogeneity, indirect interaction, and the coexistence of prey species. Am. Nat. 124, 377–406.
Holt, R. D. (1985). Population dynamics in two-patch environments: some anomalous consequences of an optimal habitat distribution. Theor. Popul. Biol. 28, 181–208.
Huang, Y. and O. Diekmann (2001). Predator migration in response to prey density: what are the consequences?. J. Math. Biol. 43, 561–581.
Huang, Y. and O. Diekmann (2003). Interspecific influence on mobility and Turing instability. Bull. Math. Biol. 46, 143–156.
Jansen, V. A. A. and A. L. Lloyd (2000). Local stability analysis of spatially homogeneous solutions of multi-patch systems. J. Math. Biol. 41, 232–252.
Karevia, P. and P. Odell (1987). Swarms of predators exhibit preytaxis if individual predators use area restricted search. Am. Nat. 130, 233–270.
Kot, M. (1992). Discrete time travelling waves: ecological examples. J. Math. Biol. 30, 413–436.
Levin, S. A. and L. A. Segel (1976). Hypothesis for origin of planktonic patchiness. Nature 259, 659.
Lewis, M. A. and S. W. Pacala (2000). Modeling and analysis of stochastic invasion processes. J. Math. Biol. 41, 387–429.
Mollison, D. (1977). Spatial contact models for ecological and epidemic spread. J. R. Stat. Soc. B 39, 283–326.
Neubert, M. G., P. Klepac and P. van den Driessche (2002). Stabilizing dispersal delays in predato-prey metapopulation models. Theor. Popul. Biol. 61, 339–347.
Neubert, M. G., M. Kot and M. A. Lewis (1995). Dispersal and pattern formation in a discrete-time predator-prey model. Theor. Popul. Biol. 48, 7–43.
Okubo, A. (1980). Diffusion and Ecological Problems: Mathematical Models, Berlin: Springer.
Okubo, A. and S. A. Levin (2001). Diffusion and Ecological Problems: Modern Perspectives, 2nd edn, Berlin: Spinger.
Ortega, J. (1987). Matrix Theory, New York, London: Plenum Press.
Othmer, H. G., S. R. Dunbar and W. Alt (1988). Models of dispersal in biological systems. J. Math. Biol. 26, 263–298.
Othmer, H. G. and L. E. Scriven (1971). Instability and dynamic patterns in cellular networks. J. Theor. Biol. 32, 507–537.
Rohani, P. and G. D. Ruxton (1999). Dispersal-induced instabilities in host-parasitoid metapopulations. Theor. Popul. Biol. 55, 23–36.
Ruxton, G. D. (1996). Density-dependent migration and stability in a system of linked populations. Bull. Math. Biol. 58, 643–660.
Segel, L. A. (1984). Taxes in cellular ecology, In Mathematical Ecology, S. A. Levin, (Ed.), Lecture Notes in Biomathematics 54, pp. 407–424.
Silva, J. A. L., M. L. De Castro and D. A. R. Justo (2001). Stability in a metapopulation model with density-dependent dispersal. Bull. Math. Biol. 63, 485–505.
Skellam, J. G. (1973). The formulation and interpretation of mathematical models of diffusionary processes in population biology, in Mathematical Theory of Dynamics of Biological Populations, M. S. Barlett and R. W. Hiorns (Eds), New York: Academic Press, pp. 63–85.
Thieme, H. R. (1979a). Asymptotic estimates of solutions of nonlinear integral equations and asymptotic speeds for the spread of populations. J. Reine Angew. Math. 306, 94–121.
Thieme, H. R. (1979b). Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread. J. Math. Biol. 8, 173–187.
Turing, A. (1952). The chemical basis of morphogenesis. Phil. Trans. R. Soc. B. 237, 37–72.
Weinberger, H. F., M. A. Lewis and B. Li (2002). Analysis of linear determinacy for spread in cooperative models. J. Math. Biol. 45, 183–218.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Huang, Y., Diekmann, O. & Van Den Bosch, F. Double-jump migration and diffusive instability. Bull. Math. Biol. 66, 487–504 (2004). https://doi.org/10.1016/j.bulm.2003.09.004
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1016/j.bulm.2003.09.004