×

Effects of a degeneracy in the competition model. I: Classical and generalized steady-state solutions. (English) Zbl 1042.35016

This paper is devoted to how heterogeneous environmental effects can greatly change the dynamical behaviour of a population model. The author considers the competition model given by \[ \begin{cases} \frac{\partial u}{\partial t}-d_1(x)\Delta u=\lambda a_1(x)u-b(x)u^2-c(x)uv\\ \frac{\partial v}{\partial t}-d_2(x)\Delta v=\mu a_2(x)v-e(x)v^2-d(x)uv\end{cases}\tag{1} \] where \(x\in\Omega\) and \(t\geq 0\), \(\Omega\) denotes a smooth bounded domain in \(\mathbb{R}^N\) \((N\geq 2)\), \(d_1,d_2,a_1,a_2,b,c,d,e\) are nonnegative functions over \(\Omega\), and \(\lambda,\mu\) are positive constants. Throughout this paper the author supposes that \(u\) and \(v\) satisfy homogeneous Dirichlet boundary conditions. The author shows that there exists a critical number \(\lambda_*\) such that (1) behaves as if \(b(x)\) is a positive constant if \(\lambda<\lambda_*\), but if \(\lambda>\lambda_*\), then interesting new phenomena occur. This shows that heterogeneous environmental effects on population models are not only quantitative, but can be qualitative as well.

MSC:

35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35B40 Asymptotic behavior of solutions to PDEs
35K57 Reaction-diffusion equations
92D25 Population dynamics (general)

Citations:

Zbl 1042.35017
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ali, S. W.; Cosner, C., Models for the effects of individual size and spatial scale on competition between species in heterogeneous environments, Math. Biosci., 127, 45-76 (1995) · Zbl 0821.92023
[2] Blat, J.; Brown, K. J., Bifurcation of steady-state solutions in predator-prey and competition systems, Proc. Roy. Soc. Edinburgh. Sect. A, 97, 21-34 (1984) · Zbl 0554.92012
[3] Blat, J.; Brown, K. J., Global bifurcation of positive solutions in some systems of elliptic equations, SIAM J. Math. Anal., 17, 1339-1353 (1986) · Zbl 0613.35008
[4] Cantrell, R. S.; Cosner, C., Should a park be an island?, SIAM J. Appl. Math., 53, 219-252 (1993) · Zbl 0811.92022
[5] Cantrell, R. S.; Cosner, C.; Hutson, V., Ecological models, permanence and spatial heterogeneity, Rocky Mountain J. Math., 26, 1-35 (1996) · Zbl 0851.92019
[6] Cosner, C.; Lazer, A. C., Stable coexistence state in the Volterra-Lotka competition model with diffusion, SIAM J. Appl. Math., 44, 1112-1132 (1984) · Zbl 0562.92012
[7] Crandall, M. G.; Rabinowitz, P. H., Bifurcation from simple eigenvalues, J. Funct. Anal., 8, 321-340 (1971) · Zbl 0219.46015
[8] Dancer, E. N., A counterexample on competing species equations, Differential Integral Equations, 9, 239-246 (1996) · Zbl 0842.35033
[9] Dancer, E. N., On positive solutions of some pairs of differential equations, II, J. Differential Equations, 60, 236-258 (1985) · Zbl 0549.35024
[10] Dancer, E. N., On the existence and uniqueness of positive solutions for competing species models with diffusion, Trans. Amer. Math. Soc., 326, 829-859 (1991) · Zbl 0769.35016
[11] Dancer, E. N.; Du, Y., Competing species equations with diffusion, large interactions, and jumping nonlinearities, J. Differential Equations, 114, 434-475 (1994) · Zbl 0815.35024
[12] Dancer, E. N.; Hess, P., Stability of fixed points for order-preserving discrete-time dynamical systems, J. Reine Angew. Math., 419, 125-139 (1991) · Zbl 0728.58018
[13] Du, Y., Uniqueness, multiplicity and stability for positive solutions of a pair of reaction-diffusion equations, Proc. Roy. Soc. Edinburgh Sect. A, 126, 777-809 (1996) · Zbl 0864.35035
[15] Du, Y.; Brown, K. J., Bifurcation and monotonicity in competition reaction-diffusion systems, Nonlinear Anal., 23, 1-13 (1994) · Zbl 0807.35041
[16] Du, Y.; Huang, Q., Blow-up solutions for a class of semilinear elliptic and parabolic equations, SIAM J. Math. Anal., 31, 1-18 (1999) · Zbl 0959.35065
[19] Du, Y.; Lou, Y., Some uniqueness and exact multiplicity results for a predator-prey model, Trans. Amer. Math. Soc., 349, 2443-2475 (1997) · Zbl 0965.35041
[20] Du, Y.; Lou, Y., S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model, J. Differential Equations, 144, 390-440 (1998) · Zbl 0970.35030
[21] Eilbeck, J.; Furter, J.; López-Gómez, J., Coexistence in a competition model with diffusion, J. Differential Equations, 107, 96-139 (1994) · Zbl 0833.92010
[22] Fraile, J. M.; Koch Medina, P.; López-Gómez, J.; Merino, S., Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic problem, J. Differential Equations, 127, 295-319 (1996) · Zbl 0860.35085
[23] Garcia-Melian, J.; Gomez-Renasco, R.; López-Gómez, J.; Sabina de Lis, J., Pointwise growth and uniqueness of positive solutions for a class of sublinear elliptic problems where bifurcation from infinity occurs, Arch. Rational Mech. Anal., 145, 261-289 (1998) · Zbl 0926.35036
[24] Gui, C.; Lou, Y., Uniqueness and nonuniqueness of coexistence states in the Lotka-Volterra competition model, Comm. Pure Appl. Math., 47, 1-24 (1994)
[25] Hess, P.; Lazer, A. C., On an abstract competition model and applications, Nonlinear Anal., 16, 917-940 (1991) · Zbl 0743.35033
[26] Hirsch, M., Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math., 383, 1-53 (1988) · Zbl 0624.58017
[27] Koman, P.; Leung, A. W., A general monotone scheme for elliptic systems with applications to ecological models, Proc. Roy. Soc. Edinburgh Sect. A, 102, 315-325 (1986) · Zbl 0606.35034
[29] López-Gómez, J., On the structure of the permanence region for competing species models with general diffusivities and transport effects, Discrete Continuous Dynam. Systems, 2, 525-542 (1996) · Zbl 0952.92029
[30] López-Gómez, J.; Sabina de Lis, J., First variation of principal eigenvalues with respect to the domain and point-wise growth of positive solutions for problems where bifurcation from infinity occurs, J. Differential Equations, 148, 47-64 (1998) · Zbl 0915.35080
[31] López-Gómez, J.; Sabina de Lis, J., Coexistence states and global attractivity for some convective diffusive competing species models, Trans. Amer. Math. Soc., 347, 3797-3833 (1995) · Zbl 0848.35012
[32] Matano, H., Existence of nontrivial unstable sets for equilibriums of strongly order-preserving systems, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 30, 645-673 (1984) · Zbl 0545.35042
[33] Ouyang, T., On the positive solutions of semilinear equations \(Δu + λu − hu^p =0\) on the compact manifolds, Trans. Amer. Math. Soc., 331, 503-527 (1992) · Zbl 0759.35021
[34] Ruan, W. H.; Pao, C. V., Positive steady-state solutions of a competing reaction-diffusion system, J. Differential Equations, 117, 401-427 (1995) · Zbl 0923.35063
[35] Smith, H. L., Monotone Dynamical Systems. Monotone Dynamical Systems, Math. Surveys and Monographs, 41 (1995), Amer. Math. Soc: Amer. Math. Soc Providence
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.