Block-pulse integrodifference equations

We present a hybrid method for calculating the equilibrium population-distributions of integrodifference equations (IDEs) with strictly increasing growth, for populations that are confined to a finite habitat-patch. This method is based on approximating the growth function of the IDE with a piecewise-constant function, and we call the resulting model a block-pulse IDE. We explicitly write out analytic expressions for the iterates and equilibria of the block-pulse IDEs as sums of cumulative distribution functions. We characterize the dynamics of one-, two-, and three-step block-pulse IDEs, including formal stability analyses, and we explore the bifurcation structure of these models. These simple models display rich dynamics, with numerous fold bifurcations. We then use three-, five-, and ten-step block-pulse IDEs, with a numerical root finder, to approximate models with compensatory Beverton-Holt growth and depensatory, or Allee-effect, growth. Our method provides a good approximation for the equilibrium distributions for compensatory and depensatory growth and offers numerical and analytical advantages over the original growth models.

Spreading Speed in an Integrodifference Predator-Prey System without Comparison Principle

In this paper, we study the spreading speed in an integrodifference system which models invasion of predators into the habitat of the prey. Without the requirement of comparison principle, we construct several auxiliary integrodifference equations and use the results of monotone scalar equations to estimate the spreading speed of the invading predators. We also present some numerical simulations to support our theoretical results and demonstrate that the integrodifference predator-prey system exhibits very complex dynamics. Our theory and numerical results imply that the invasion of predators may have a rough constant speed. Moreover, our numerical simulations indicate that the spatial contact of individuals and the overcompensatory phenomenon of the prey may be conducive to the persistence of nonmonotone biological systems and lead to instability of the predator-free state.

Modeling the dispersal-reproduction trade-off in an expanding population

Trade-offs between dispersal and reproduction are known to be important drivers of population dynamics, but their direct influence on the spreading speed of a population is not well understood. Using integrodifference equations, we develop a model that incorporates a dispersal-reproduction trade-off which allows for a variety of different shaped trade-off curves. We show there is a unique reproductive-dispersal allocation that gives the largest value for the spreading speed and calculate the sensitivities of the reproduction, dispersal, and trade-off shape parameters. Uncertainty in the model parameters affects the expected spread of the population and we calculate the optimal allocation of resources to dispersal that maximizes the expected spreading speed. Higher allocation to dispersal arises from uncertainty in the reproduction parameter or the shape of the reproduction trade-off curve. Lower allocation to dispersal arises from uncertainty in the shape of the dispersal trade-off curve, but does not come from uncertainty in the dispersal parameter. Our findings give insight into how parameter sensitivity and uncertainty influence the spreading speed of a population with a dispersal-reproduction trade-off.

Inside Dynamics of Integrodifference Equations with Mutations

The method of inside dynamics provides a theory that can track the dynamics of neutral gene fractions in spreading populations. However, the role of mutations has so far been absent in the study of the gene flow of neutral fractions via inside dynamics. Using integrodifference equations, we develop a neutral genetic mutation model by extending a previously established scalar inside dynamics model. To classify the mutation dynamics, we define a mutation class as the set of neutral fractions that can mutate into one another. We show that the spread of neutral genetic fractions is dependent on the leading edge of population as well as the structure of the mutation matrix. Specifically, we show that the neutral fractions that contribute to the spread of the population must belong to the same mutation class as the neutral fraction found in the leading edge of the population. We prove that the asymptotic proportion of individuals at the leading edge of the population spread is given by the dominant right eigenvector of the associated mutation matrix, independent of growth and dispersal parameters. In addition, we provide numerical simulations to demonstrate our mathematical results, to extend their generality and to develop new conjectures about our model.

Invasion speeds of Triatoma dimidiata, vector of Chagas disease: An application of orthogonal polynomials method

Demographic processes and spatial dispersal of Triatoma dimidiata, a triatomine species vector of Chagas disease, are modeled by integrodifference equations to estimate invasion capacity of this species under different ecological conditions. The application of the theory of orthogonal polynomials and the steepest descent method applied to these equations, allow a good approximation of the abundance of the adult female population and the invasion speed. We show that: (1) under the same mean conditions of demography and dispersal, periodic spatial dispersal results in an invasion speed 2.5 times larger than the invasion speed when spatial dispersal is continuous; (2) when the invasion speed of periodic spatial dispersal is correlated to adverse demographic conditions, it is 34.7% higher as compared to a periodic dispersal that is correlated to good demographic conditions. From our results we conclude, in terms of triatomine population control, that the invasive success of T. dimidiata may be most sensitive to the probability of transition from juvenile to adult stage. We discuss our main theoretical predictions in the light of observed data in different triatomines species found in the literature.

Approximating the Critical Domain Size of Integrodifference Equations

Integrodifference (IDE) models can be used to determine the critical domain size required for persistence of populations with distinct dispersal and growth phases. Using this modelling framework, we develop a novel spatially implicit approximation to the proportion of individuals lost to unfavourable habitat outside of a finite domain of favourable habitat, which consistently outperforms the most common approximations. We explore how results using this approximation compare to the existing IDE results on the critical domain size for populations in a single patch of good habitat, in a network of patches, in the presence of advection, and in structured populations. We find that the approximation consistently provides results which are in close agreement with those of an IDE model except in the face of strong advective forces, with the advantage of requiring fewer numerical approximations while providing insights into the significance of disperser retention in determining the critical domain size of an IDE.