ON THE EFFECTS OF DENSITY-DEPENDENT EMIGRATION ON ECOLOGICAL MODELS WITH LOGISTIC AND WEAK ALLEE TYPE GROWTH TERMS

We analyze the structure of positive steady states for a population model designed to explore the effects of habitat fragmentation, density dependent emigration, and Allee effect growth. The steady state reaction diffusion equation is: { -Delta u = lambda f(u); Omega partial derivative u/partial derivative eta + gamma root lambda g(u)u =0; partial derivative Omega where f(s) = 1/a s(1-s) (a + s ) can represent either logistic-type growth (a >= 1) or weak Allee affect growth (a is an element of (0, 1)), lambda, gamma > 0 are parameters, Omega is a bounded domain in R-N; N > 1 with smooth boundary partial derivative Omega or Omega = (0; 1), partial derivative/partial derivative eta is the outward normal derivative of u, and g(u) is related to the relationship between density and emigration. In particular, we consider three forms of emigration: density independent emigration (g = 1), a negative density dependent emigration of the form g(s) = 1/1+beta s, and a positive density dependent emigration of the form g(s) = 1 + beta s, where beta > 0 is a parameter representing the interaction strength. We establish existence, nonexistence, and multiplicity results for ranges of lambda depending on the choice of the function g. Our existence and multiplicity results are proved via the method of sub-super-solutions and study of certain eigenvalue problems. For the case Omega = (0, 1); we also provide exact bifurcation diagrams for positive solutions for certain values of the parameters alpha, beta and gamma via a quadrature method and Mathematica computations. Our results shed light on the complex interactions of density dependent mechanisms on population dynamics in the presence of habitat fragmentation.