Spreading speeds of spatially periodic integro-difference models for populations with nonmonotone recruitment functions. (English) Zbl 1141.92041
Summary: An idea used by H. R. Thieme [J. Math. Biol. 8, 173–187 (1979; Zbl 0417.92022)] is extended to show that a class of integro-difference models for a periodically varying habitat has a spreading speed and a formula for it is given, even when the recruitment function \(R(u, x)\) is not nondecreasing in \(u\), so that overcompensation occurs. Numerical simulations illustrate the behavior of the solutions of the recursion whose initial values vanish outside a bounded set.
MSC:
| 92D40 | Ecology |
| 35K55 | Nonlinear parabolic equations |
| 45M99 | Qualitative behavior of solutions to integral equations |
| 92D25 | Population dynamics (general) |
| 35K57 | Reaction-diffusion equations |
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
Citations:
Zbl 0417.92022References:
| [1] | Berestycki, H., Hamel, F.: Front propagation in periodic excitable media. Commun. Pure Appl. Math. 55, 949–1032 (2002) · Zbl 1024.37054 · doi:10.1002/cpa.3022 |
| [2] | Berestycky, H., Hamel, F., Roques, L.: Analysis of the periodically fragmented environment model: II–Biological invasions and pulsating travelling fronts. J. Math. Pures Appl. 84, 1101–1146 (2005) · Zbl 1083.92036 · doi:10.1016/j.matpur.2004.10.006 |
| [3] | Cantrell, R.S., Cosner, C.: Spatial Ecology via Reaction–Diffusion Systems. Wiley, New York (2003) · Zbl 1059.92051 |
| [4] | Courant, R., Hilbert, D.: Methods of Mathematical Physics II. Interscience, New York (1962) · Zbl 0099.29504 |
| [5] | Diekmann, O.: Run for your life. J. Differ. Equ. 33, 58–73 (1979) · doi:10.1016/0022-0396(79)90080-9 |
| [6] | Freidlin, M.I.:On wavefront propagation in periodic media. In: Pinsky, M. (ed.) Stochastic Analysis and Applications. Adv. in Prob. and Rel. Topics 7, Marcel Dekker, New York (1984) · Zbl 0549.60054 |
| [7] | Gärtner, J., Freidlin, M.I.: On the propagation of concentration waves in periodic and random media. Sov. Math. Dokl. 20, 1282–1286 (1979) · Zbl 0447.60060 |
| [8] | Hess, P., Kato, T.: On some linear and nonlinear eigenvalue problems with an indefinite weight function. Commun. Partial Differ. Equ. 5, 999–1030 (1980) · Zbl 0477.35075 · doi:10.1080/03605308008820162 |
| [9] | Kawasaki, K., Shigesada, N.: An integrodifference model for biological invasions in a periodically fragmented environment. Jpn. J. Ind. App. Math. 24, 3–15 (2007) · Zbl 1116.92066 · doi:10.1007/BF03167504 |
| [10] | Kinezaki, N., Kawasaki, K., Shigesada, N.: Spatial dynamics of invasion in sinusoidally varying environments. Popul. Ecol. 48, 263–270 (2006) · doi:10.1007/s10144-006-0263-2 |
| [11] | Kinezaki, N., Kawasaki, K., Takasu, F., Shigesada, N.: Modeling biological invasions into periodically fragmented environments. Theor. Popul. Biol. 64, 291–302 (2003) · Zbl 1102.92057 · doi:10.1016/S0040-5809(03)00091-1 |
| [12] | Kot, M.: Discrete-time travelling waves: ecological examples. J. Math. Biol. 30, 413–436 (1992) · Zbl 0825.92126 · doi:10.1007/BF00173295 |
| [13] | Kot, M., Lewis, M.A., van den Driessche, P.: Dispersal data and the spread of invading organisma. Ecology 77, 2027–2042 (1996) · doi:10.2307/2265698 |
| [14] | Lax, P.D.: Functional Analysis. Wiley-Interscience, New York (2002) · Zbl 1009.47001 |
| [15] | Lutscher, F., Lewis, M.A., McCauley, E.: Effects of heterogeneity on spread and persistence in rivers. Bull. Math. Biol. 68, 2129–2160 (2006) · Zbl 1296.92211 · doi:10.1007/s11538-006-9100-1 |
| [16] | May, R.M., Oster, G.F.: Period doubling and the onset of turbulence: an analytic estimate of the Feigenbaum ratio. Phys. Lett. 78, 1–3 (1980) · doi:10.1016/0375-9601(80)90788-4 |
| [17] | Mollison, D.: The rate of spatial propagation of simple epidemics. Proc. Sixth Berkeley Symp. Math. Statist. Probab. 3, 579–614 (1972) · Zbl 0266.92008 |
| [18] | Papanicolaou, G., Xin, J.: Reaction–diffusion fronts in periodically layered media. J. Stat. Phys. 63, 915–932 (1991) · doi:10.1007/BF01029991 |
| [19] | Robbins, T.C., Lewis, M.A.: Modeling population spread in heterogeneous environments using integrodifference equations. Preprint (2006) |
| [20] | Roques, L., Stoica, R.S.: Species persistence decreases with habitat fragmentation: an analysis in periodic stochastic environments. J. Math. Biol. 55, 189–205 (2007) · Zbl 1128.35361 · doi:10.1007/s00285-007-0076-8 |
| [21] | Shigesada, N., Kawasaki, K.: Biological Invasions: Theory and Practice. Oxford University Press, Oxford (1997) |
| [22] | Shigesada, N., Kawasaki, K., Teramoto, E.: Traveling periodic waves in heterogeneous environments. Theor. Popul. Biol. 30, 143–160 (1986) · Zbl 0591.92026 · doi:10.1016/0040-5809(86)90029-8 |
| [23] | Thieme, H.R.: Asymptotic estimates of the solutions of nonlinear integral equations and asymptotic speeds for spread of populations. J. für reine u. Angew. Math. 306, 94–121 (1979) · Zbl 0395.45010 · doi:10.1515/crll.1979.306.94 |
| [24] | Thieme, H.R.: Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread. J. Math. Biol. 8, 173–187 (1979) · Zbl 0417.92022 · doi:10.1007/BF00279720 |
| [25] | Thieme, H.R., Zhao, X.Q.: Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction–diffusion models. J. Differ. Equ. 195, 430–470 (2003) · Zbl 1045.45009 · doi:10.1016/S0022-0396(03)00175-X |
| [26] | Van Kirk, R.W., Lewis, M.A.: Integrodifference models for persistence in fragmented habitats. Bull. Math. Biol. 59, 107–137 (1997) · Zbl 0923.92031 · doi:10.1007/BF02459473 |
| [27] | Weinberger, H.F.: Asymptotic behavior of a model in population genetics. In: Chadam, J.M. (ed.) Nonlinear Partial Differential Equations and Applications. Lecture Notes in Mathematics, vol. 648, pp. 47–96. Springer, Berlin (1978) |
| [28] | Weinberger, H.F.: Long-time behavior of a class of biological models. SIAM J. Math. Anal. 13, 353–396 (1982) · Zbl 0529.92010 · doi:10.1137/0513028 |
| [29] | Weinberger, H.F.: On spreading speeds and traveling waves for growth and migration models in a periodic habitat. J. Math. Biol. 45, 511–548 (2002) · Zbl 1058.92036 · doi:10.1007/s00285-002-0169-3 |
| [30] | Xin, J.: Analysis and modeling of front propagation in heterogeneous media. SIAM Rev. 42, 161–230 (2000) · Zbl 0951.35060 · doi:10.1137/S0036144599364296 |
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