On allee effects in structured populations

Maps f(x) = A( x) x of the nonnegative cone C of R-k into itself are considered where A( x) are nonnegative, primitive matrices with nondecreasing entries and at least one increasing entry. Let lambda( x) denote the dominant eigenvalue of A(x) and lambda(infinity) = sup(xis an element ofC) lambda(x). These maps are shown to exhibit a dynamical trichotomy. First, if lambda(0) greater than or equal to 1, then lim(n-->infinity)parallel tof(n)(x)parallel to = infinity for all nonzero x is an element of C. Second, if lambda(infinity) less than or equal to 1, then lim(n-->infinity) f(n)( x) = 0 for all x is an element of C. Finally, if lambda(0) < 1 and lambda(infinity) > 1, then there exists a compact invariant hypersurface Gamma separating C. For x below Gamma, lim(n-->infinity) f(n)( x) = 0, while for x above, lim(n-->infinity)parallel tof(n)(x)parallel to = infinity. An application to nonlinear Leslie matrices is given.