Turing patterns in a predator–prey model with seasonality

2018

J. Math. Biol.

Vegetation patterns are a characteristic feature of semi-arid regions. On hillsides these patterns occur as stripes running parallel to the contours. The Klausmeier model, a coupled reaction-advection-diffusion system, is a deliberately simple model describing the phenomenon. In this paper, we replace the diffusion term describing plant dispersal by a more realistic nonlocal convolution integral to account for the possibility of long-range dispersal of seeds. Our analysis focuses on the rainfall level at which there is a transition between uniform vegetation and pattern formation. We obtain results, valid to leading order in the large parameter comparing the rate of water flow downhill to the rate of plant dispersal, for a negative exponential dispersal kernel. Our results indicate that both a wider dispersal of seeds and an increase in dispersal rate inhibit the formation of patterns. Assuming an evolutionary trade-off between these two quantities, mathematically motivated by the limiting behaviour of the convolution term, allows us to make comparisons to existing results for the original reaction-advection-diffusion system. These comparisons show that the nonlocal model always predicts a larger parameter region supporting pattern formation. We then numerically extend the results to other dispersal kernels, showing that the tendency to form patterns depends on the type of decay of the kernel.

Section: Discussionmentioning

confidence: 99%

Analysis of a model for banded vegetation patterns in semi-arid environments with nonlocal dispersal

2018

J. Math. Biol.

“…This property is specific to the system considered in this paper and no generalisations can be made. Indeed, diffusion-driven instabilities have been shown to occur in other impulsive models [97].…”

Section: Discussionmentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

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

Physica D: Nonlinear Phenomena

2020

Patterns of vegetation are a characteristic feature of many semi-arid regions. The limiting resource in these ecosystems is water, which is added to the system through short and intense rainfall events that cause a pulse of biological processes such as plant growth and seed dispersal. We propose an impulsive model based on the Klausmeier reaction-advection-diffusion system, analytically investigate the effects of rainfall intermittency on the onset of patterns, and augment our results by numerical simulations of model extensions. Our investigation focuses on the parameter region in which a transition between uniform and patterned vegetation occurs. Results show that decay-type processes associated with a low frequency of precipitation pulses inhibit the onset of patterns and that under intermittent rainfall regimes, a spatially uniform solution is sustained at lower total precipitation volumes than under continuous rainfall, if plant species are unable to efficiently use low soil moisture levels. Unlike in the classical setting of a reaction-diffusion model, patterns are not caused by a diffusion-driven instability but by a combination of sufficiently long periods of droughts between precipitation pulses and water diffusion. Our results further indicate that the introduction of pulse-type seed dispersal weakens the effects of changes to width and shape of the plant dispersal kernel on the onset of patterns. MSC codes: 35R09, 35R12, 35B36

“…In separate work, we account for a combination of rainfall, plant growth and seed dispersal pulses with the continuous nature of plant loss and water evaporation and drainage, using an impulsive model (Eigentler and Sherratt 2020 ). Such models combine partial differential equations with integrodifference equations (see for example Wang and Lutscher ( 2018 ) for an impulsive model in the context of predator-prey dynamics with synchronised predator reproduction). The impulsive model has its own limitations as it can only take into account a periodic separation of precipitation events, but not any seasonal patterns.…”

Section: Discussionmentioning

confidence: 99%

An integrodifference model for vegetation patterns in semi-arid environments with seasonality

2020

J. Math. Biol.

Vegetation patterns are a characteristic feature of semi-deserts occurring on all continents except Antarctica. In some semi-arid regions, the climate is characterised by seasonality, which yields a synchronisation of seed dispersal with the dry season or the beginning of the wet season. We reformulate the Klausmeier model, a reaction–advection–diffusion system that describes the plant–water dynamics in semi-arid environments, as an integrodifference model to account for the temporal separation of plant growth processes during the wet season and seed dispersal processes during the dry season. The model further accounts for nonlocal processes involved in the dispersal of seeds. Our analysis focusses on the onset of spatial patterns. The Klausmeier partial differential equations (PDE) model is linked to the integrodifference model in an appropriate limit, which yields a control parameter for the temporal separation of seed dispersal events. We find that the conditions for pattern onset in the integrodifference model are equivalent to those for the continuous PDE model and hence independent of the time between seed dispersal events. We thus conclude that in the context of seed dispersal, a PDE model provides a sufficiently accurate description, even if the environment is seasonal. This emphasises the validity of results that have previously been obtained for the PDE model. Further, we numerically investigate the effects of changes to seed dispersal behaviour on the onset of patterns. We find that long-range seed dispersal inhibits the formation of spatial patterns and that the seed dispersal kernel’s decay at infinity is a significant regulator of patterning.