On diffusive population models with toxicants and time delays. (English) Zbl 0927.35049

Feng, Wei; Lu, Xin

J. Math. Anal. Appl. 233, No. 1, 373-386 (1999).

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.

Reviewer: I.Onciulescu (Iaşi)

MSC:

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

Keywords:

unique positive steady-state solution

References:

[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: 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: 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: Academic Press New York · Zbl 0777.34002