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Loops and branches of coexistence states in a Lotka–Volterra competition model. (English) Zbl 1154.35011
Summary: A two-species Lotka–Volterra competition–diffusion model with spatially inhomogeneous reaction terms is investigated. The two species are assumed to be identical except for their interspecific competition coefficients. Viewing their common diffusion rate \(\mu\) as a parameter, we describe the bifurcation diagram of the steady states, including stability, in terms of two real functions of \(\mu\). We also show that the bifurcation diagram can be rather complicated. Namely, given any two positive integers \(l\) and \(b\), the interspecific competition coefficients can be chosen such that there exist at least \(l\) bifurcating branches of positive stable steady states which connect two semi-trivial steady states of the same type (they vanish at the same component), and at least \(b\) other bifurcating branches of positive stable steady states that connect semi-trivial steady states of different types.

MSC:
35B40 Asymptotic behavior of solutions to PDEs
35K57 Reaction-diffusion equations
92D10 Genetics and epigenetics
92D25 Population dynamics (general)
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[1] Abraham, J.; Robbin, R., Transversal mappings and flows, (1967), Benjamin New York · Zbl 0171.44404
[2] Cantrell, R.S.; Cosner, C., The effects of spatial heterogeneity in population dynamics, J. math. biol., 29, 315-338, (1991) · Zbl 0722.92018
[3] Cantrell, R.S.; Cosner, C., Should a park be an island?, SIAM J. appl. math., 53, 219-252, (1993) · Zbl 0811.92022
[4] Cantrell, R.S.; Cosner, C., On the effects of spatial heterogeneity on the persistence of interacting species, J. math. biol., 37, 103-145, (1998) · Zbl 0948.92021
[5] Cantrell, R.S.; Cosner, C., Spatial ecology via reaction – diffusion equations, Ser. math. comput. biology, (2003), Wiley Chichester, UK · Zbl 1059.92051
[6] Cantrell, R.S.; Cosner, C.; Hutson, V., Permanence in ecological systems with diffusion, Proc. roy. soc. Edinburgh sect. A, 123, 553-559, (1993) · Zbl 0796.92026
[7] Cantrell, R.S.; Cosner, C.; Hutson, V., Ecological models, permanence and spatial heterogeneity, Rocky mountain J. math., 26, 1-35, (1996) · Zbl 0851.92019
[8] Cantrell, R.S.; Cosner, C.; Lou, Y., Multiple reversals of competitive dominance in ecological reserves via external habitat degradation, J. dynam. differential equations, 16, 973-1010, (2004) · Zbl 1065.35140
[9] Dockery, J.; Hutson, V.; Mischaikow, K.; Pernarowski, M., The evolution of slow dispersal rates: A reaction – diffusion model, J. math. biol., 37, 61-83, (1998) · Zbl 0921.92021
[10] Du, Y., Effects of a degeneracy in the competition model, part II. perturbation and dynamical behavior, J. differential equations, 181, 133-164, (2002) · Zbl 1042.35017
[11] Du, Y., Realization of prescribed patterns in the competition model, J. differential equations, 193, 147-179, (2003) · Zbl 1274.35137
[12] Furter, J.E.; López-Gómez, J., Diffusion-mediated permanence problem for a heterogeneous lotka – volterra competition model, Proc. roy. soc. Edinburgh sect. A, 127, 281-336, (1997) · Zbl 0941.92022
[13] Gilbarg, D.; Trudinger, N., Elliptic partial differential equation of second order, (1983), Springer Berlin
[14] Hastings, A., Spatial heterogeneity and ecological models, Ecology, 71, 426-428, (1990)
[15] Henry, D., Geometric theory of semilinear parabolic equations, (1981), Springer New York · Zbl 0456.35001
[16] Henry, D., Perturbation of the boundary for boundary value problems of partial differential operators, (2005), Cambridge Univ. Press Cambridge
[17] Hess, P., Periodic parabolic boundary value problems and positivity, (1991), Longman Scientific & Technical Harlow, UK · Zbl 0731.35050
[18] Holmes, E.E.; Lewis, M.A.; Banks, J.E.; Veit, R.R., Partial differential equations in ecology: spatial interactions and population dynamics, Ecology, 75, 17-29, (1994)
[19] Hsu, S.; Smith, H.; Waltman, P., Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. amer. math. soc., 348, 4083-4094, (1996) · Zbl 0860.47033
[20] Hutson, V.; López-Gómez, J.; Mischaikow, K.; Vickers, G., Limit behavior for a competing species problem with diffusion, (), 501-533 · Zbl 0848.35057
[21] Hutson, V.; Lou, Y.; Mischaikow, K., Spatial heterogeneity of resources versus lotka – volterra dynamics, J. differential equations, 185, 97-136, (2002) · Zbl 1023.35047
[22] Hutson, V.; Lou, Y.; Mischaikow, K., Convergence in competition models with small diffusion coefficients, J. differential equations, 211, 135-161, (2005) · Zbl 1074.35054
[23] Hutson, V.; Lou, Y.; Mischaikow, K.; Poláčik, P., Competing species near the degenerate limit, SIAM J. math. anal., 35, 453-491, (2003) · Zbl 1043.35080
[24] Hutson, V.; Martinez, S.; Mischaikow, K.; Vickers, G.T., The evolution of dispersal, J. math. biol., 47, 483-517, (2003) · Zbl 1052.92042
[25] Hutson, V.; Mischaikow, K.; Poláčik, P., The evolution of dispersal rates in a heterogeneous time-periodic environment, J. math. biol., 43, 501-533, (2001) · Zbl 0996.92035
[26] Kato, T., Perturbation theory for linear operators, (1966), Springer New York · Zbl 0148.12601
[27] López-Gómez, J., Coexistence and meta-coexistence for competing species, Houston J. math., 29, 483-536, (2003) · Zbl 1034.35062
[28] López-Gómez, J.; Molina-Meyer, M., Superlinear indefinite system beyond lotka – volterra models, J. differential equations, 221, 343-411, (2006) · Zbl 1093.35033
[29] Lunardi, A., Analytic semigroups and optimal regularity in parabolic problems, (1995), Birkhäuser Berlin · Zbl 0816.35001
[30] Pacala, S.; Roughgarden, J., Spatial heterogeneity and interspecific competition, Theor. pop. biol., 21, 92-113, (1982) · Zbl 0492.92017
[31] Potapov, A.B.; Lewis, M.A., Climate and competition: the effect of moving range boundaries on habitat invasibility, Bull. math. biol., 975-1008, (2004) · Zbl 1334.92454
[32] Protter, M.H.; Weinberger, H.F., Maximum principles in differential equations, (1984), Springer Berlin · Zbl 0153.13602
[33] Shigesada, N.; Kawasaki, K., Biological invasions: theory and practice, Oxford series in ecology and evolution, (1997), Oxford Univ. Press Oxford
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