zbMATH — the first resource for mathematics

On diffusive population models with toxicants and time delays. (English) Zbl 0927.35049
The authors study two generalized reaction-diffusion systems, as logistic systems in mathematical ecology. The first of these is the following: \[ \partial u(t,x)/\partial t- Au(t,x)= r(x) u(t,x)(K(x)- u(t,x))/(K(x)+ c(x)u(t,x))\text{ in }[0,\infty)\times \Omega, \] \[ B[u](t,x)= 0\quad\text{on }(0,\infty)\times \partial\Omega,\quad u(0,x)= u_0(x)\quad\text{on }\overline\Omega, \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) with smooth boundary \(\partial\Omega\); the functions \(r\), \(c\) and \(K\) are positive in \(\Omega\) and Hölder continuous on \(\overline\Omega\); \(B[u]= u\) or \(B[u]= \partial u/\partial\nu+ \gamma(x)u\), with \(\gamma\in C^{1+\alpha}(\partial\Omega)\) and \(\gamma(x)\geq 0\) on \(\partial\Omega\); the initial function \(u_0\) is a Hölder continuous function on \(\overline\Omega\), and the differential operator \(A\) is a uniformly strongly elliptic operator.
The main goal of this paper is to show the existence of a unique positive steady-state solution in these models and investigate the asymptotic behavior of the time-dependent solutions, in both models, in relation to such steady solutions.

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions to PDEs
92D40 Ecology
Full Text: DOI
[1] Black, J.; Brown, K.J., Bifurcation of steady-state solutions in predator-prey and competition systems, Proc. roy. soc. Edinburgh sect. A, 97, 21-34, (1989) · Zbl 0554.92012
[2] Feng, W.; Lu, X., Asymptotic periodicity in diffusive logistic equations with discrete delays, Nonlinear anal., 26, 171-178, (1996) · Zbl 0842.35129
[3] Gopalsamy, K.; Kulenovic, M.R.S.; Ladas, G., Time lags in a “food-limited” population model, Appl. anal., 31, 225-237, (1988) · Zbl 0639.34070
[4] Gopalsamy, K.; Kulenovic, M.R.S.; Ladas, G., Environmental periodicity and time delays in a “food-limited” population model, J. math. anal. appl., 147, 545-555, (1990) · Zbl 0701.92021
[5] Hallam, T.G.; DeLuna, J.T., Effects of toxicants on populations: A qualitative approach III, J. theor. biol., 109, 411-429, (1984)
[6] Lu, X., Monotone method and convergence acceleration for finite-difference solutions of parabolic problems with time delays, Numer. methods partial differential equations, 11, 591-602, (1995) · Zbl 0839.65096
[7] Lu, X.; Feng, W., Periodic solution and oscillation in a competition model with diffusion and distributed delay effects, Nonlinear anal., 27, 699-709, (1996) · Zbl 0862.35134
[8] Pao, C.V., On nonlinear parabolic and elliptic equations, (1992), Plenum Press New York · Zbl 0780.35044
[9] Pao, C.V., Numerical methods for semilinear parabolic equations, SIAM J. numer. anal., 24, 24-35, (1987) · Zbl 0623.65100
[10] Pielou, E.C., An introduction to mathematical ecology, (1969), Wiley New York · Zbl 0259.92001
[11] Smith, F.E., Population dynamics in daphnia magna, Ecology, 44, 651-663, (1963)
[12] Kuang, Y., Delay differential equations with applications in population dynamics, Mathematics in science and engineering, (1993), Academic Press New York · Zbl 0777.34002
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.