Dynamics from a predator-prey-quarry-resource-scavenger model

Modelling the Impact of NETosis During the Initial Stage of Systemic Lupus Erythematosus

The development of autoimmune diseases often takes years before clinical symptoms become detectable. We propose a mathematical model for the immune response during the initial stage of Systemic Lupus Erythematosus which models the process of aberrant apoptosis and activation of macrophages and neutrophils. NETosis is a type of cell death characterised by the release of neutrophil extracellular traps, or NETs, containing material from the neutrophil's nucleus, in response to a pathogenic stimulus. This process is hypothesised to contribute to the development of autoimmunogenicity in SLE. The aim of this work is to study how NETosis contributes to the establishment of persistent autoantigen production by analysing the steady states and the asymptotic dynamics of the model by numerical experiment.

Effect of scavenging on predation in a food web

Scavenging can have important consequences for food web dynamics, for example, it may support additional consumer species and affect predation on live prey. Still, few food web models include scavenging. We develop a dynamic model that includes two facultative scavenger species, which we refer to as the predator or scavenger species according to their natural scavenging propensity, as well as live prey, and a carrion pool to show ramifications of scavenging for predation in simple food webs. Our modeling suggests that the presence of scavengers can both increase and decrease predator kill rates and overall predation in model food webs and the impact varies (in magnitude and direction) with context. In particular, we explore the impact of the amount of dynamics (exploitative competition) allowed in the predator, scavenger, and prey populations as well as the direction and magnitude of interference competition between predators and scavengers. One fundamental prediction is that scavengers most likely increase predator kill rates, especially if there are exploitative feedback effects on the prey or carrion resources like is normally observed in natural systems. Scavengers only have minimal effects on predator kill rate when predator, scavenger, and prey abundances are kept constant by management. In such controlled systems, interference competition can greatly affect the interactions in contrast to more natural systems, with an increase in interference competition leading to a decrease in predator kill rate. Our study adds to studies that show that the presence of predators affects scavenger behavior, vital rates, and food web structure, by showing that scavengers impact predator kill rates through multiple mechanisms, and therefore indicating that scavenging and predation patterns are tightly intertwined. We provide a road map to the different theoretical outcomes and their support from different empirical studies on vertebrate guilds to provide guidance in wildlife management.

Dynamical behavior of a delayed prey-predator-scavenger system with fear effect and linear harvesting

In this paper, we propose a delayed prey-predator-scavenger system with fear effect and linear harvesting. First, we discuss the existence and stability of all possible equilibria. Next, we investigate the existence of Hopf bifurcation of the delayed system by regarding the gestation period of the scavenger as a bifurcation parameter. Furthermore, we obtain the direction of Hopf bifurcation and the stability of bifurcating periodic solutions by using the normal form theory and the central manifold theorem. In addition, we give the optimal harvesting strategy of the delayed system based on Pontryagin’s maximum principle with delay. Finally, some numerical simulations are carried out to verify our theoretical results.

A Non-local Cross-Diffusion Model of Population Dynamics I: Emergent Spatial and Spatiotemporal Patterns

We extend a spatially non-local cross-diffusion model of aggregation between multiple species with directed motion toward resource gradients to include many species and more general kinds of dispersal. We first consider diffusive instabilities, determining that for directed motion along fecundity gradients, the model permits the Turing instability leading to colony formation and persistence provided there are three or more interacting species. We also prove that such patterning is not possible in the model under the Turing mechanism for two species under directed motion along fecundity gradients, confirming earlier findings in the literature. However, when the directed motion is not along fecundity gradients, for instance, if foraging or migration is sub-optimal relative to fecundity gradients, we find that very different colony structures can emerge. This generalization also permits colony formation for two interacting species. In the advection-dominated case, aggregation patterns are more broad and global in nature, due to the inherent non-local nature of the advection which permits directed motion over greater distances, whereas in the diffusion-dominated case, more highly localized patterns and colonies develop, owing to the localized nature of random diffusion. We also consider the interplay between Turing patterning and spatial heterogeneity in resources. We find that for small spatial variations, there will be a combination of Turing patterns and patterning due to spatial forcing from the resources, whereas for large resource variations, spatial or spatiotemporal patterning can be modified greatly from what is predicted on homogeneous domains. For each of these emergent behaviors, we outline the theoretical mechanism leading to colony formation and then provide numerical simulations to illustrate the results. We also discuss implications this model has for studies of directed motion in different ecological settings.

Generalist predator dynamics under kolmogorov versus non-Kolmogorov models

Ecosystems often contain multiple species across two or more trophic levels, with a variety of interactions possible. In this paper we study two classes of models for generalist predators that utilize more than one food source. These models fall into two categories: predator - two prey and predator - prey - subsidy models. For the former, we consider a generalist predator which utilizes two distinct prey species, modelled via a Kolmogorov system of equations with Type II response functions. For the latter, we consider a generalist predator which exploits both a prey population and an allochthonous resource which is provided as a subsidy to the system exogenously, again with Type II response functions. This latter class of model is no longer Kolmogorov in form, due to an exogenous forcing term modelling the input of the allochthonous resource into the system. We non-dimensionalize both models, so that their respective parameter spaces may be more easily compared, and study the dynamics possible from each type of model, which will then indicate - for specific parameter regimes - which generalist predator's preferences are more favorable to survival, including the prevalence of coexistence states. We also consider the various non-equilibrium dynamics emergent from such models, and show that the non-Kolmogorov predator - prey - subsidy model of 10 admits more regular dynamics (including steady states and one type of limit cycle), whereas the predator - two prey Kolmogorov model can feature multiple types of limit cycles, as well as multistability resulting in strong sensitivity to initial conditions (with stable limit cycles and steady states both coexisting for the same model parameters). Our results highlight several interesting differences and similarities between Kolmogorov and non-Kolmogorov models for generalist predators.

Turing Instability and Colony Formation in Spatially Extended Rosenzweig-MacArthur Predator-Prey Models with Allochthonous Resources

While it is somewhat well known that spatial PDE extensions of the Rosenzweig-MacArthur predator-prey model do not admit spatial pattern formation through the Turing mechanism, in this paper we demonstrate that the addition of allochthonous resources into the system can result in spatial patterning and colony formation. We study pattern formation, through Turing and Turing-Hopf mechanisms, in two distinct spatial Rosenzweig-MacArthur models generalized to include allochthonous resources. Both models have previously been shown to admit heterogeneous spatial solutions when prey and allochthonous resources are confined to different regions of the domain, with the predator able to move between the regions. However, pattern formation in such cases is not due to the Turing mechanism, but rather due to the spatial separation between the two resources for the predator. On the other hand, for a variety of applications, a predator can forage over a region where more than one food source is present, and this is the case we study in the present paper. We first consider a three PDE model, consisting of equations for each of a predator, a prey, and an allochthonous resource or subsidy, with all three present over the spatial domain. The second model we consider arises in the study of two independent predator-prey systems in which a portion of the prey in the first system becomes an allochthonous resource for the second system; this is referred to as a predator-prey-quarry-resource-scavenger model. We show that there exist parameter regimes for which these systems admit Turing and Turing-Hopf bifurcations, again resulting in spatial or spatiotemporal patterning and hence colony formation. This is interesting from a modeling standpoint, as the standard spatially extended Rosenzweig-MacArthur predator-prey equations do not permit the Turing instability, and hence, the inclusion of allochthonous resources is one route to realizing colony formation under Rosenzweig-MacArthur kinetics. Concerning the ecological application, we find that spatial patterning occurs when the predator is far more mobile than the prey (reflected in the relative difference between their diffusion parameters), with the prey forming colonies and the predators more uniformly dispersed throughout the domain. We discuss how this spatially heterogeneous patterning, particularly of prey populations, may constitute one way in which the paradox of enrichment is resolved in spatial systems by way of introducing allochthonous resource subsidies in conjunction with spatial diffusion of predator and prey populations.

A stochastic differential game approach toward animal migration

A stochastic differential game model for animal migration between two habitats under uncertain environment, a new population dynamics model, is formulated. Its novelty is the use of an impulse control formalism to naturally describe migrations with different timings and magnitudes that the conventional models could not handle. Uncertainty of the environment that the population faces with is formulated in the context of the multiplier robust control. The optimal migration strategy to give the maximized minimal profit is found through a Hamilton-Jacobi-Bellman quasi-variational inequality (HJBQVI). A key message from HJBQVI is that its free boundary determines the optimal migration strategy. Solving the HJBQVI is carried out with a specialized stable and convergent finite difference scheme. This paper theoretically suggests that the sub-additivity of the performance index, the index to be optimized through the migration, critically affects the resulting strategy. The computational results with the established scheme are consistent with the theoretical predictions and support importance of the sub-additivity property. Social interaction to reduce the net mortality rate is also quantified, suggesting a linkage between the present and existing population dynamics models.

Hybrid approach to modeling spatial dynamics of systems with generalist predators

We consider hybrid spatial modeling approaches for ecological systems with a generalist predator utilizing a prey and either a second prey or an allochthonous resource. While spatial dispersion of populations is often modeled via stepping-stone (discrete spatial patches) or continuum (one connected spatial domain) formulations, we shall be interested in hybrid approaches which we use to reduce the dimension of certain components of the spatial domain, obtaining either a continuum model of varying spatial dimensions, or a mixed stepping-stone-continuum model. This approach results in models consisting of partial differential equations for some of the species which are coupled via reactive boundary conditions to lower dimensional partial differential equations or ordinary differential equations for the other species. In order to demonstrate the use of this approach, we consider two case studies. In the first case study, we consider a one-predator two-prey interaction between beavers, wolves and white-tailed deer in Voyageurs National Park. In the second case study, we consider predator-prey-allochthonous resource interactions between bears, berries and salmon on Kodiak Island. For each case study, we compare the results from the hybrid modeling approach with corresponding stepping-stone and continuum model results, highlighting benefits and limitations of the method. In some cases, we find that the hybrid modeling approach allows for solutions which are easier to simulate (akin to stepping-stone models) while maintaining seemingly more realistic spatial dynamics (akin to full continuum models).