Front propagation in the shadow wave-pinning model.

In this paper we consider a non-local bistable reaction-diffusion equation which is a simplified version of the wave-pinning model of cell polarization. In the small diffusion limit, a typical solution u(x, t) of this model approaches one of the stable states of the bistable nonlinearity in different parts of the spatial domain [Formula: see text], separated by an interface moving at a normal velocity regulated by the integral [Formula: see text]. In what is often referred to as wave-pinning, feedback between mass-conservation and bistablity causes the interface to slow and approach a fixed limit. In the limit of a small diffusivity [Formula: see text], we prove that for any [Formula: see text] the interface can be estimated within [Formula: see text] of the location as predicted using formal asymptotics. We also discuss the sharpness of our result by comparing the formal asymptotic results with numerical simulations.

Niche differentiation in the light spectrum promotes coexistence of phytoplankton species: a spatial modelling approach.

The paradox of the plankton highlights the apparent contradiction between Gause's law of competitive exclusion and the observed diversity of phytoplankton. It is well known that phytoplankton dynamics depend heavily on light availability. Here we treat light as a continuum of resources rather than a single resource by considering the visible light spectrum. We propose a spatially explicit reaction-diffusion-advection model to explore under what circumstance coexistence is possible from mathematical and biological perspectives. Furthermore, we provide biological context as to when coexistence is expected based on the degree of niche differentiation within the light spectrum and overall turbidity of the water.

A cancer model with nonlocal free boundary dynamics.

Cancer cells at the tumor boundary move in the direction of the oxygen gradient, while cancer cells far within the tumor are in a necrotic state. This paper introduces a simple mathematical model that accounts for these facts. The model consists of cancer cells, cytotoxic T cells, and oxygen satisfying a system of partial differential equations. Some of the model parameters represent the effect of anti-cancer drugs. The tumor boundary is a free boundary whose dynamics is determined by the movement of cancer cells at the boundary. The model is simulated for radially symmetric and axially symmetric tumors, and it is shown that the tumor may increase or decrease in size, depending on the "strength" of the drugs. Existence theorems are proved, global in-time in the radially symmetric case, and local in-time for any shape of tumor. In the radially symmetric case, it is proved, under different conditions, that the tumor may shrink monotonically, or expand monotonically.

Directed movement changes coexistence outcomes in heterogeneous environments.

Understanding mechanisms of coexistence is a central topic in ecology. Mathematical analysis of models of competition between two identical species moving at different rates of symmetric diffusion in heterogeneous environments show that the slower mover excludes the faster one. The models have not been tested empirically and lack inclusions of a component of directed movement toward favourable areas. To address these gaps, we extended previous theory by explicitly including exploitable resource dynamics and directed movement. We tested the mathematical results experimentally using laboratory populations of the nematode worm, Caenorhabditis elegans. Our results not only support the previous theory that the species diffusing at a slower rate prevails in heterogeneous environments but also reveal that moderate levels of a directed movement component on top of the diffusive movement allow species to coexist. Our results broaden the theory of species coexistence in heterogeneous space and provide empirical confirmation of the mathematical predictions.

Three-patch Models for the Evolution of Dispersal in Advective Environments: Varying Drift and Network Topology.

We study the evolution of dispersal in advective three-patch models with distinct network topologies. Organisms can move between connected patches freely and they are also subject to the passive, directed drift. The carrying capacity is assumed to be the same in all patches, while the drift rates could vary. We first show that if all drift rates are the same, the faster dispersal rate is selected for all three models. For general drift rates, we show that the infinite diffusion rate is a local Convergence Stable Strategy (CvSS) for all three models. However, there are notable differences for three models: For Model I, the faster dispersal is always favored, irrespective of the drift rates, and thus the infinity dispersal rate is a global CvSS. In contrast, for Models II and III a singular strategy will exist for some ranges of drift rates and bi-stability phenomenon happens, i.e., both infinity and zero diffusion rates are local CvSSs. Furthermore, for both Models II and III, it is possible for two competing populations to coexist by varying drift and diffusion rates. Some predictions on the dynamics of n-patch models in advective environments are given along with some numerical evidence.

Analysis of a mathematical model of immune response to fungal infection.

Fungi are cells found as commensal residents, on the skin, and on mucosal surfaces of the human body, including the digestive track and urogenital track, but some species are pathogenic. Fungal infection may spread into deep-seated organs causing life-threatening infection, especially in immune-compromised individuals. Effective defense against fungal infection requires a coordinated response by the innate and adaptive immune systems. In the present paper we introduce a simple mathematical model of immune response to fungal infection consisting of three partial differential equations, for the populations of fungi (F), neutrophils (N) and cytotoxic T cells (T), taking N and T to represent, respectively, the innate and adaptive immune cells. We denote by [Formula: see text] the aggressive proliferation rate of the fungi, by [Formula: see text] and [Formula: see text] the killing rates of fungi by neutrophils and T cells, and by [Formula: see text] and [Formula: see text] the immune strengths, respectively, of N and T of an infected individual. We take the expression [Formula: see text] to represent the coordinated defense of the immune system against fungal infection. We use mathematical analysis to prove the following: If [Formula: see text], then the infection is eventually stopped, and [Formula: see text] as [Formula: see text]; and (ii) if [Formula: see text] then the infection cannot be stopped and F converges to some positive constant as [Formula: see text]. Treatments of fungal infection include anti-fungal agents and immunotherapy drugs, and both cause the parameter I to increase.

Are Two-Patch Models Sufficient? The Evolution of Dispersal and Topology of River Network Modules.

We study the dynamics of two competing species in three-patch models and illustrate how the topology of directed river network modules may affect the evolution of dispersal. Each model assumes that patch 1 is at the upstream end, patch 3 is at the downstream end, but patch 2 could be upstream, or middle stream, or downstream, depending on the specific topology of the modules. We posit that individuals are subject to both unbiased dispersal between patches and passive drift from one patch to another, depending upon the connectivity of patches. When the drift rate is small, we show that for all models, the mutant species can invade when rare if and only if it is the slower disperser. However, when the drift rate is large, most models predict that the faster disperser wins, while some predict that there exists one evolutionarily singular strategy. The intermediate range of drift is much more complex: most models predict the existence of one singular strategy, but it may or may not be evolutionarily stable, again depending upon the topology of modules, while one model even predicts that for some intermediate drift rate, singular strategy does not exist and the faster disperser wins the competition.

Analysis of a mathematical model of rheumatoid arthritis.

Rheumatoid arthritis is an autoimmune disease characterized by inflammation in the synovial fluid within the synovial joint connecting two contiguous bony surfaces. The inflammation diffuses into the cartilage adjacent to each of the bony surfaces, resulting in their gradual destruction. The interface between the cartilage and the synovial fluid is an evolving free boundary. In this paper we consider a two-phase free boundary problem based on a simplified model of rheumatoid arthritis. We prove global existence and uniqueness of a solution, and derive properties of the free boundary. In particular it is proved that the free boundary increases in time, and the cartilage shrinks to zero as [Formula: see text], even under treatment by a drug. It is also shown in the reduced one-phased problem, with cartilage alone, that a larger prescribed inflammation function leads to a faster destruction of the cartilage.

Dynamics of a consumer-resource reaction-diffusion model : Homogeneous versus heterogeneous environments.

We study the dynamics of a consumer-resource reaction-diffusion model, proposed recently by Zhang et al. (Ecol Lett 20(9):1118-1128, 2017), in both homogeneous and heterogeneous environments. For homogeneous environments we establish the global stability of constant steady states. For heterogeneous environments we study the existence and stability of positive steady states and the persistence of time-dependent solutions. Our results illustrate that for heterogeneous environments there are some parameter regions in which the resources are only partially limited in space, a unique feature which does not occur in homogeneous environments. Such difference between homogeneous and heterogeneous environments seems to be closely connected with a recent finding by Zhang et al. (2017), which says that in consumer-resource models, homogeneously distributed resources could support higher population abundance than heterogeneously distributed resources. This is opposite to the prediction by Lou (J Differ Equ 223(2):400-426, 2006. https://doi.org/10.1016/j.jde.2005.05.010 ) for logistic-type models. For both small and high yield rates, we also show that when a consumer exists in a region with a heterogeneously distributed input of exploitable renewed limiting resources, the total population abundance at equilibrium can reach a greater abundance when it diffuses than when it does not. In contrast, such phenomenon may fail for intermediate yield rates.

Single species growth consuming inorganic carbon with internal storage in a poorly mixed habitat.

This paper presents a PDE system modeling the growth of a single species population consuming inorganic carbon that is stored internally in a poorly mixed habitat. Inorganic carbon takes the forms of "CO2" (dissolved CO2 and carbonic acid) and "CARB" (bicarbonate and carbonate ions), which are substitutable in their effects on algal growth. We first establish a threshold type result on the extinction/persistence of the species in terms of the sign of a principal eigenvalue associated with a nonlinear eigenvalue problem. If the habitat is the unstirred chemostat, we add biologically relevant assumptions on the uptake functions and prove the uniqueness and global attractivity of the positive steady state when the species persists.

Dimorphism by Singularity Theory in a Model for River Ecology.

Geritz, Gyllenberg, Jacobs, and Parvinen show that two similar species can coexist only if their strategies are in a sector of parameter space near a nondegenerate evolutionarily singular strategy. We show that the dimorphism region can be more general by using the unfolding theory of Wang and Golubitsky near a degenerate evolutionarily singular strategy. Specifically, we use a PDE model of river species as an example of this approach. Our finding shows that the dimorphism region can exhibit various different forms that are strikingly different from previously known results in adaptive dynamics.

Evolution of dispersal in closed advective environments.

We study a two-species competition model in a closed advective environment, where individuals are exposed to unidirectional flow (advection) but no individuals are lost through the boundary. The two species have the same growth and advection rates but different random dispersal rates. The linear stability analysis of the semi-trivial steady state suggests that, in contrast to the case without advection, slow dispersal is generally selected against in closed advective environments. We investigate the invasion exponent for various types of resource functions, and our analysis suggests that there might exist some intermediate dispersal rate that will be selected. When the diffusion and advection rates are small and comparable, we determine criteria for the existence and multiplicity of singular strategies and evolutionarily stable strategies. We further show that every singular strategy is convergent stable.

Evolutionarily stable and convergent stable strategies in reaction-diffusion models for conditional dispersal.

We consider a mathematical model of two competing species for the evolution of conditional dispersal in a spatially varying, but temporally constant environment. Two species are different only in their dispersal strategies, which are a combination of random dispersal and biased movement upward along the resource gradient. In the absence of biased movement or advection, Hastings showed that the mutant can invade when rare if and only if it has smaller random dispersal rate than the resident. When there is a small amount of biased movement or advection, we show that there is a positive random dispersal rate that is both locally evolutionarily stable and convergent stable. Our analysis of the model suggests that a balanced combination of random and biased movement might be a better habitat selection strategy for populations.

Evolution of conditional dispersal: evolutionarily stable strategies in spatial models.

We consider a two-species competition model in which the species have the same population dynamics but different dispersal strategies. Both species disperse by a combination of random diffusion and advection along environmental gradients, with the same random dispersal rates but different advection coefficients. Regarding these advection coefficients as movement strategies of the species, we investigate their course of evolution. By applying invasion analysis we find that if the spatial environmental variation is less than a critical value, there is a unique evolutionarily singular strategy, which is also evolutionarily stable. If the spatial environmental variation exceeds the critical value, there can be three or more evolutionarily singular strategies, one of which is not evolutionarily stable. Our results suggest that the evolution of conditional dispersal of organisms depends upon the spatial heterogeneity of the environment in a subtle way.