zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 $\bbfR^n$ 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 $\gamma(x)\ge 0$ 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.

MSC:
 35K60 Nonlinear initial value problems for linear parabolic equations 35K57 Reaction-diffusion equations 35B40 Asymptotic behavior of solutions of PDE 92D40 Ecology
Full Text:
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) [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) · Zbl 0777.35001 [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) · 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. (1993) · Zbl 0777.34002