In this paper, we analyse Turing instability and bifurcations in a host–parasitoid model with nonlocal effect. For a ordinary differential equation model, we provide some preliminary analysis on Hopf bifurcation. For a reaction–diffusion model with local intraspecific prey competition, we first explore the Turing instability of spatially homogeneous steady states. Next, we show that the model can undergo Hopf bifurcation and Turing–Hopf bifurcation, and find that a pair of spatially nonhomogeneous periodic solutions is stable for a (8,0)-mode Turing–Hopf bifurcation and unstable for a (3,0)-mode Turing–Hopf bifurcation. For a reaction–diffusion model with nonlocal intraspecific prey competition, we study the existence of the Hopf bifurcation, double-Hopf bifurcation, Turing bifurcation, and Turing–Hopf bifurcation successively, and find that a spatially nonhomogeneous quasi-periodic solution is unstable for a (0,1)-mode double-Hopf bifurcation. Our results indicate that the model exhibits complex pattern formations, including transient states, monostability, bistability, and tristability. Finally, numerical simulations are provided to illustrate complex dynamics and verify our theoretical results.

When faced with slowly depleting resources (such as decrease in precipitation due to climate change), complex ecological systems are prone to sudden irreversible changes (such as desertification) as the resource level dips below a tipping point of the system. A possible coping mechanism is the formation of spatial patterns, which allows for concentration of sparse resources and the survival of the species within “ecological niches” even below the tipping point of the homogeneous vegetation state. However, if the change in resource availability is too sudden, the system may not have time to transition to the patterned state and will pass through the tipping point instead, leading to extinction. We argue that the deciding factors are the speed of resource depletion and the amount of the background noise (seasonal climate changes) in the system. We illustrate this phenomenon on a model of patterned vegetation. Our analysis underscores the importance of, and the interplay between, the speed of climate change, heterogeneity of the environment, and the amount of seasonal variability.

The nonlocal Fisher equation has been proposed as a simple model exhibiting Turinginstability and the interpretation refers to adaptive evolution. By analogy with other formalismsused in adaptive dynamics, it is expected that concentration phenomena (like convergence to a sumof Dirac masses) will happen in the limit of small mutations. In the present work we study thisasymptotics by using a change of variables that leads to a constrained Hamilton-Jacobi equation.We prove the convergence analytically and illustrate it numerically. We also illustrate numericallyhow the constraint is related to the concentration points. We investigate numerically some featuresof these concentration points such as their weights and their numbers. We show analytically howthe constrained Hamilton-Jacobi gives the so-called canonical equation relating their motion withthe selection gradient. We illustrate this point numerically.

We present a three species model describing the degradation of substrate bytwo competing populations of microorganisms in a marine sediment. Considering diffusionto be the main transport process, we obtain a reaction diffusion system (RDS) which westudy in terms of spontaneous pattern formation. We find that the conditions for patternsto evolve are likely to be fulfilled in the sediment. Additionally, we present simulationsthat are consistent with experimental data from the literature. We focus on the interactionbetween pattern formation and non-uniform forcing, which naturally occurs in our model,since layers closer to the sea water are richer in particular chemicals than deeper layers. We find that heterogeneous forcing has a strong impact on the structures that evolve, leading tothe formation of banded structures or even pulses instead of the well-known Turing patterns.