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Discrete-time growth-dispersal models. (English) Zbl 0595.92011
A nonlinear integrodifference equation is presented as a model of a population with growth and dispersal in discrete time. The equation takes the form \[ N_{t+1}(x)=\int^{b}_{a}k(x,y)f(N_ t(y))dy. \] The population is viewed as synchronous, with nonoverlapping generations, and having two distinct phases, one sedentary and one dispersing. \(N_ t(x)\) is the population at spatial coordinate x at the start of the \(t^{th}\) sedentary stage and k(x,y)dy is the probability that an individual moves during its dispersal stage from the interval \((y,y+dy]\) to x. In an isotropic environment the kernel k is symmetric and various examples of such kernels are considered.
The stability of equilibria is discussed by means of the method of linearization. A number of examples are given to illustrate various properties of the solutions such as bifurcation of nontrivial equilibria, period-doubling, chaos, and diffusive instability.
Reviewer: G.F.Webb

MSC:
92D25 Population dynamics (general)
45M10 Stability theory for integral equations
45H05 Integral equations with miscellaneous special kernels
45J05 Integro-ordinary differential equations
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[1] Aikman, D.; Hewitt, G., An experimental investigation of the rate and form of dispersal in grasshoppers, J. appl. ecol., 9, 807-817, (1972)
[2] Aronson, D.G.; Chory, M.A.; Hall, G.R.; McGehee, R.P., Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study, Comm. math. phys., 83, 303-354, (1982) · Zbl 0499.70034
[3] Casten, R.; Holland, C., Instability results for reaction-diffusion equations with Neumann boundary conditions, J. differential equations, 27, 266-273, (1978) · Zbl 0338.35055
[4] Dobzhansky, T.; Wright, S., Genetics of natural populations, Genetics, 28, 304-340, (1943), X. Dispersion rates in Drosophila Pseudoobscura
[5] Dobzhansky, T.; Wright, S., Genetics of natural populations, Genetics, 32, 303-324, (1947), XV. Rate of diffusion of a mutant gene through a population of Drosophila Pseudoobscura
[6] Duffy, E., Aerial dispersal in a known spider population, J. anim. ecol., 25, 85-111, (1956)
[7] Feigenbaum, M., Quantitative universality for a class of nonlinear transformations, J. statist. phys., 19, 25-52, (1978) · Zbl 0509.58037
[8] Feigenbaum, M., Universal behavior in nonlinear systems, Los alamos sci., 1, 4-27, (1980)
[9] Fife, P.; Peletier, L.A., Clines induced by variable selection and migration, Proc. roy. soc. London ser. B, 214, 99-123, (1981)
[10] Fisher, R.A., The wave of advance of advantageous genes, Ann. eugen. London, 7, 355-369, (1937) · JFM 63.1111.04
[11] Fusco, G.; Hale, J.K., Stable equilibria in a scalar parabolic equation with variable diffusion, SIAM J. math. anal., 16, 1152-1164, (1985) · Zbl 0597.35040
[12] Gadgil, M., Dispersal: population consequences and evolution, Ecology, 52, 253-261, (1971)
[13] Gillespie, J.H., The role of migration in the genetic structure of populations in temporally and spatially varying environments, Amer. natur., 117, 223-233, (1981), III. Migration modification
[14] Greenberg, J.M.; Hastings, S.P., Spatial patterns for discrete models of diffusion in excitable media, SIAM J. appl. math., 34, 515-523, (1978) · Zbl 0398.92004
[15] Guckenheimer, J.; Oster, G.F.; Ipaktchi, A., The dynamics of density dependent population models, J. math. biol., 4, 101-147, (1976) · Zbl 0379.92016
[16] J. Hale and J. Vegas, A nonlinear parabolic equation with varying domain, LCDS Report 81-1, Brown Univ. · Zbl 0569.35048
[17] Hamilton, W.D.; May, R.M., Dispersal in stable habitats, Nature, 269, 578-581, (1977)
[18] Hammerstein, A., Nichtlineare integralgleichungen nebst anwendungen, Acta math., 54, 117-176, (1930) · JFM 56.0343.03
[19] Hassell, M.P., The dynamics of arthropod predator-prey systems, (1978), Princeton U.P Princeton, N.J · Zbl 0429.92018
[20] Hutson, V.; Pym, J.S., Applications of functional analysis and operator theory, (1980), Academic London · Zbl 0426.46009
[21] Johnson, C.G., Migration and dispersal of insects by flight, (1969), Methuen, London
[22] Jury, E.I., Theory and application of the Z-transform method, (1964), Wiley New York
[23] Jury, E.I., Inners and stability of dynamical systems, (1974), Wiley New York · Zbl 0307.93025
[24] Kaneko, K., Transition from torus to chaos accompanied by frequency lockings with symmetry breaking, Progr. theoret. phys., 69, 1427-1442, (1983) · Zbl 1200.37032
[25] Karlin, S.; McGregor, J., Polymorphisms for genetic and ecological systems with weak coupling, Theoret. population biol., 3, 210-238, (1972) · Zbl 0262.92007
[26] Kirstead, H.; Slobodkin, L.B., The size of water masses containing plankton Bloom, J. mar. res., 12, 141-147, (1953)
[27] Kloeden, P.E., Cycles and chaos in higher dimensional difference equations, Proceedings of the ixth international conference on nonlinear oscillations, (1981), Kiev · Zbl 0471.39001
[28] Kolmogorov, A.; Petrovsky, I.; Piscounov, N., Étude de l’équation de las diffusion avec croissance de la quantité de matière et son application a un problème biologique, Moscow univ. bull. ser. internat. sect. A, 1, 1-25, (1937)
[29] Kot, M.; Schaffer, W.M., The effects of seasonality on discrete models of population growth, Theoret. population biol., 26, 340-360, (1984) · Zbl 0551.92014
[30] Krasnoselskii, M.A., Topological methods in the theory of nonlinear integral equations, (1964), Pergamon Oxford
[31] Krasnoselskii, M.A.; Zabreiko, P.P., Geometrical methods of nonlinear analysis, (1984), Springer Berlin
[32] Kuramoto, Y., Diffusion-induced chemical turbulence, (), 134 · Zbl 0435.92030
[33] Levin, S., Dispersion and population interactions, Amer. natur., 108, 207-228, (1974)
[34] Levin, S., Non-uniform stable solutions to reaction-diffusion equations: applications to ecological pattern formation, (), 210-222
[35] Levin, S.; Goodyear, C.P., Analysis of an age-structured fishery model, J. math. biol., 9, 245-274, (1980) · Zbl 0424.92020
[36] Levin, S.; Segel, L.A., Pattern generation in space and aspect, SIAM rev., 27, 45-67, (1985) · Zbl 0576.92008
[37] R. McMurtie, Persistence and stability of single-species and prey-predator systems in spatially heterogeneous environments, Math. Biosci. 39:11-51.
[38] Marotto, F.R., Snap-back repellers imply chaos in Rn, J. math. anal. appl., 63, 199-223, (1978) · Zbl 0381.58004
[39] Marotto, F.R., The dynamics of a discrete population model with threshold, Math. biosci., 58, 123-128, (1982) · Zbl 0486.92017
[40] Matano, H., Asymptotic behavior and stability of semi-linear diffusion equations, Publ. res. inst. math. sci., 15, 401-451, (1979)
[41] May, R.M., On relationships among various types of population models, Amer. natur., 107, 46-57, (1972)
[42] May, R.M., Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos, Science, 186, 645-647, (1974)
[43] May, R.M., Biological populations obeying difference equations: stable points, stable cycles, and chaos, J. theoret. biol., 49, 511-524, (1975)
[44] May, R.M., Nonlinear problems in ecology and resource management, Lecture notes for LES houches summer school on chaotic behavior of deterministic systems, (1981)
[45] May, R.M., Regulation of populations with nonoverlapping generations by microparasites: A purely chaotic system, Amer. natur., 125, 573-584, (1985)
[46] May, R.M.; Oster, G.F., Bifurcations and dynamic complexity in simple ecological models, Amer. natur., 110, 573-599, (1976)
[47] Maynard-Smith, J., Mathematical ideas in biology, (1968), Cambridge U.P Cambridge
[48] Mimura, M.; Nishiura, N.; Yamaguti, M., Some diffusive prey and predator systems and their bifurcation problems, Ann. New York acad. sci., 316, 490-510, (1979)
[49] Myskis, A.D., Advanced mathematics for engineers, (1975), Mir Moscow
[50] Okubo, A., Diffusion and ecological problems: mathematical models, (1980), Springer Berlin · Zbl 0422.92025
[51] L.F. Olsen and H. Degn, Chaos in biological systems, Quart. Rev. Biophys., to appear.
[52] Ricker, W.E., Stock and recruitment, J. fish. res. board. canad., 11, 559-623, (1954)
[53] Rodgers, T.D., Chaos in systems in population biology, Progr. theoret. biol., 6, 91-146, (1981)
[54] Rodgers, T.D., Rarity and chaos, Math. biosci., 72, 13-17, (1984) · Zbl 0556.92017
[55] Sarkovskii, A.N., Coexistence of cycles of a continuous map of a line into itself, Ukrain. mat. zh., 16, 61-71, (1964)
[56] Schaffer, W.M.; Kot, M., Nearly one dimensional dynamics in an epidemic, J. theoret. biol., 112, 403-427, (1985)
[57] Schaffer, W.M.; Kot, M., Do strange attractors govern ecological systems?, Bioscience, 35, 342-350, (1985)
[58] W.M. Schaffer and M. Kot, Differential systems in ecology and epidemiology, in Chaos: An Introduction (A.V. Holden, Ed.), Univ. of Manchester Press, to appear.
[59] Segel, L.A.; Jackson, J.L., Dissipative structure: an explanation and an ecological example, J. theoret. biol., 37, 545-559, (1972)
[60] Segel, L.A.; Levin, S.A., Applications of nonlinear stability theory to the study of the effects of dispersion on predator-prey interactions, (), 123-152
[61] Skellam, J.G., Random dispersal in theoretical populations, Biometrika, 38, 196-218, (1951) · Zbl 0043.14401
[62] Smoller, J., Shock waves and reaction-diffusion equations, (1983), Springer New York · Zbl 0508.35002
[63] Smoller, J.; Wasserman, A., Global bifurcation of steady-state solutions, J. differential equations, 39, 269-290, (1981) · Zbl 0425.34028
[64] Stakgold, I., Green’s functions and boundary value problems, (1979), Wiley New York · Zbl 0421.34027
[65] Turing, A.M., The chemical basis of morphogenesis, Philos. trans. roy. soc. London ser. B, 237, 37-72, (1952) · Zbl 1403.92034
[66] Vance, R.R., The effect of dispersal on population size in temporally varying environment, Theoret. population biol., 18, 343-362, (1980) · Zbl 0463.92018
[67] Vance, R.R., The effect of dispersal on population stability in one-species, discrete-space population growth models, Amer. natur., 123, 230-254, (1984)
[68] Vandermeer, J., To be rare is to be chaotic, Ecology, 63, 1167-1168, (1982)
[69] Vandermeer, J., On the resolution of chaos, Theoret. population biol., 22, 17-27, (1982) · Zbl 0502.92013
[70] Weinberger, H.F., Asymptotic behavior of a model of population genetics, (), 47-98, Lecture Notes in Mathematics · Zbl 0383.35034
[71] Weinberger, H.F., Long-time behavior of a class of biological models, () · Zbl 0529.92010
[72] Wolfenbarger, D.O., Dispersion of small organisms, Amer. midland. natur., 35, 1-152, (1946)
[73] Wolfenbarger, D.O., Dispersion of small organisms, incidence of viruses and pollen; dispersion of fungus, spores, and insects, Lloydia, 22, 1-106, (1959)
[74] Wolfenbarger, D.O., Factors affecting dispersal distances of small organisms, (1975), Exposition Press Hicksville, N.Y
[75] Wolfram, S., Cellular automata, Los alamos sci., 3-21, (1983)
[76] Wolfram, S., Cellular automata as models of complexity, Nature, 311, 419-424, (1984)
[77] Zabreyko, P.P.; Koshelev, A.I.; Krasnosel’skii, M.A.; Mikhlin, S.G.; Rakovshchik, L.S.; Stet’senko, V.Ya., Integral equations—a reference text, (1975), Noordhoff Leyden · Zbl 0293.45001
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