Sebastian J. Schreiber

Importance of Metapopulation Connectivity to Restocking and Restoration of Marine Species

We examine the impact of spatial processes on the efficacy of restocking in species with varying forms of population or metapopulation structure. Metapopulations are classified based on spatial complexity and the degree of connectedness between populations. Designation of effective restocking sites requires careful attention to metapopulation dynamics; populations in the metapopulation can differ dramatically in demography and connectivity, and the sites they occupy can vary in habitat quality. Source populations, which are optimal for restocking, can be distinct geographically and may be a small percentage of the metapopulation. Sink areas, where restocking is almost certain to be fruitless, can nonetheless serve as productive locations for habitat restoration since larvae from source reefs are likely to recruit to these areas. Effective restocking of metapopulations is most likely to be attained by selection of optimal source populations; inattention to metapopulation dynamics can doom restoration efforts with marine species.

Handling time promotes the coevolution of aggregation in predator-prey systems

Predators often have type II functional responses and live in environments where their life history traits as well as those of their prey vary from patch to patch. To understand how spatial heterogeneity and predator handling times influence the coevolution of patch preferences and ecological stability, we perform an ecological and evolutionary analysis of a Nicholson–Bailey type model. We prove that coevolutionarily stable prey and searching predators prefer patches that in isolation support higher prey and searching predator densities, respectively. Using this fact, we determine how environmental variation and predator handling times influence the spatial patterns of patch preferences, population abundances and per-capita predation rates. In particular, long predator handling times are shown to result in the coevolution of predator and prey aggregation. An analytic expression characterizing ecological stability of the coevolved populations is derived. This expression implies that contrary to traditional theoretical expectations, predator handling time can stabilize predator–prey interactions through its coevolutionary influence on patch preferences. These results are shown to have important implications for classical biological control.

Coexistence for species sharing a predator

A class of equations describing the dynamics of two prey sharing a common predator are considered. Even though the boundary and internal dynamics can exhibit oscillatory behavior, it is shown these equations are permanent if only if they admit a positive equilibrium. Going beyond permanence, a subclass of equations are constructed that are almost surely permanent but not permanent; there exists an attractor in the positive orthant that attracts Lebesgue almost every (but not every) initial condition.

To persist or not to persist

Ecological vector fields u xi = xifi(x) on the non-negative cone R n on R n are often used to describe the dynamics of n interacting species. These vector fields are called permanent (or uniformly persistent) if the boundary ∂R n of the nonnegative cone is repelling. We construct an open set of ecological vector fields containing a dense subset of permanent vector fields and containing a dense subset of vector fields with attractors on ∂R n +. In particular, this construction implies that robustly permanent vector fields are not dense in the space of permanent vector fields. Hence, verifying robust permanence is important. We illustrate this result with ecological vector fields involving five species that admit a heteroclinic cycle between two equilibria and the Hastings–Powell teacup attractor.

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 A(x) denote the dominant eigenvalue of A(x) and λ(∞) = sup x ∈ C A(x). These maps are shown to exhibit a dynamical trichotomy. First, if A(0) > 1, then lim n→ ∞ ∥f n (x)∥ = ∞ for all nonzero x ∈ C. Second, if λ(∞) 1, then there exists a compact invariant hypersurface Γ separating C. For x below Γ, lim n→ ∞ f n (x) = 0, while for x above, lim n→ ∞ ∥f n (x)∥ = ∞. An application to nonlinear Leslie matrices is given.

Periodicity, persistence, and collapse in host–parasitoid systems with egg limitation

There is an emerging consensus that parasitoids are limited by the number of eggs which they can lay as well as the amount of time they can search for their hosts. Since egg limitation tends to destabilize host–parasitoid dynamics, successful control of insect pests by parasitoids requires additional stabilizing mechanisms such as heterogeneity in the distribution of parasitoid attacks and host density-dependence. To better understand how egg limitation, search limitation, heterogeneity in parasitoid attacks, and host density-dependence influence host–parasitoid dynamics, discrete time models accounting for these factors are analyzed. When parasitoids are purely egg-limited, a complete anaylsis of the host–parasitoid dynamics are possible. The analysis implies that the parasitoid can invade the host system only if the parasitoid’s intrinsic fitness exceeds the host’s intrinsic fitness. When the parasitoid can invade, there is a critical threshold, CV *>1, of the coefficient of variation (CV) of the distribution of parasitoid attacks that determines that outcome of the invasion. If parasitoid attacks sufficiently aggregated (i.e., CV>CV *), then the host and parasitoid coexist. Typically (in a topological sense), this coexistence is shown to occur about a periodic attractor or a stable equilibrium. If the parasitoid attacks are sufficiently random (i.e. CV<CV *), then the parasitoid drives the host to extinction. When parasitoids are weakly search-limited as well as egg-limited, coexistence about a global attractor occurs even if CV<CV *. However, numerical simulations suggest that the nature of this attractor depends critically on whether CV<1 or CV>1. When CV<1, the parasitoid exhibits highly oscillatory dynamics. Alternatively, when parasitoid attacks are sufficiently aggregated but not overly aggregated (i.e. CV>1 but close to 1), the host and parasitoid coexist about a stable equilibrium with low host densities. The implications of these results for classical biological control are discussed.