A Hybrid Continuous/Discrete-Time Model for Invasion Dynamics of Zebra Mussels in Rivers

Abstract

Abstract. While some species spread upstream in river environments, not all invasive species are successful in spreading upriver. Here the dynamics of unidirectional water flow found in rivers can play a role in determining invasion success. We develop a continuous-discrete hybrid benthic-drift population model to describe the dynamics of invasive freshwater mussels in rivers. In the model, a reaction-advection-diffusion equation coupled to an ordinary differential equation describes the larval dispersal in the drift until settling to the benthos, while two difference equations describe the population growth on the benthos. We study the population persistence criteria based on three related measures: fundamental niche, source-sink distribution, and net reproductive rate. We calculate the critical domain size in a bounded domain by analyzing a next generation operator. We analyze the upstream and downstream spreading speeds in an unbounded domain. The model is parameterized by available data in the literature. Combining the results of model parameterization and theoretical analysis, we numerically analyze how the interaction between population growth and dispersal, river flow rate, and water temperature affects both persistence and the spread of zebra mussels along a river.

FKPP equation with impulses on unbounded domain

“…Actually, there exist several possibilities including the formalism presented in system (4) (see also Lakshmikantham et al (1989) [18], Bainov and Simeonov (1995) [4], Dumont and Tchuenche (2012) [10], Dufourd and Dumont (2013) [9], Tchuinté Tamen et al (2016) [35], (2017) [36], [47] and references therein) as well as the formalism advocated by Lewis and Li (2012) [19] (see also Vasilyeva et al (2012) [42], Fazly et al (2017) [11], Huang et al (2017) [14] and references therein). For τ-periodic impulsive harvest events, the inter-harvest saison (i.e.…”

“…Homogeneous equilibria of system (6) are equilibria of model (14). Model (14) always has the trivial equilibrium e 0 = 0. Recall that…”

“…A direct comparison allows us to observe that the positive equilibrium of model (13) given by (14) corresponds to the value of the positive periodic solution (17) computed at t = (n + 1)τ, i.e. V ((n + 1)τ).…”

“…Readers are referred to Yatat et al (2017b) [46] for travelling waves in savanna reaction-diffusion models with time-continuous fire events. To the best of authors' knowledge, existence of travelling wave solutions for system of impulsive reaction-diffusion equations is not well documented except in Huang et al (2017) [14]. The rest of the paper is organized as follows.…”

Effects of precipitation intermittency on vegetation patterns in semi-arid landscapes

“…The use of impulsive models is a relatively new approach in mathematical modelling but such models are suitable for the description of a wide range of systems. Previous applications include descriptions of populations whose life cycle consists of two non-overlapping stages, such as organisms whose larvae are subjected to a water flow [95,39]; predator prey systems in which consumer reproduction occurs only once a year and is based on the amount of stored energy accumulated through consumption of prey during the year [97] or that are periodically subjected to external inputs [1]; and more general consumer-resource systems in which the consumer reproduction is synchronised [63,49] or in which seasonal harvesting occurs [49]. Impulsive models can further provide a mechanistic interpretation of the underlying ecological processes involved in purely discrete systems [29].…”