We consider two Lotka-Volterra type models for ecological communities exhibiting a modified form of cyclic competition. The first governs a four-species ecological system. When each species competes only with one other species cyclically, then, it is known that, for strong competition the system evolves to one of two alliances of non-competing species. We consider the case when there is internal competition and predation within one of the alliances. This leads to an embedded three-species rock-paper-scissors (RPS) community, which can be dynamically unstable - leading to attracting heteroclinic cycles within the four-species system. We show that, even for vanishingly small fracturing, other outcomes are also possible. We identify all possible physical steady states, their stabilities and bifurcations which can occur between them (eight bifurcations in all). For our second model, we consider the case of two ecological communities within a heterogeneous environment. The communities are coupled, allowing information exchange between them. We prove that for strong coupling the two communities will forma unified state corresponding to one with averaged ecological parameters. For the case of dynamically unstable communities (i.e., heteroclinic cycles), we develop a method to characterize the averaged heteroclinic cycle based on the rate of expansion of trajectory visits to the appropriate saddle. Information exchange can allow small ecological heterogeneities to lead to very major changes in the steady state. In particular, information exchange can quench heteroclinic cycles within both communities or conversely can allow for heteroclinic cycles where none would occur for each community in isolation.

We consider a four-species cyclic community with species ui where i=1,…,4. For each i, species ui exhibit a crowding effect, competing with members of its own species (intraspecies competition). Furthermore, there is interspecies competition as ui competes with ui+1 (mod 4). We also allow for species mobility via Fickian diffusion. A feature of this cyclic competition scheme is that there are two alliances of non-competing species, d13, consisting only of species u1 and u3, and d24, consisting only of species u2 and u4. We focus on the parameter regime where these two alliances are the only stable states. We consider the invasion problem, where these two states are initially adjacent, and derive conditions where one state displaces the other. We focus primarily on the standstill problem, where both alliances are evenly matched, dividing the parameter space into two regions - one where d13 displaces d24 and one where d24 displaces d13. We concentrate on three specific parameter regimes: (i) moderate competition - the interspecies and intraspecies competitions are roughly of equal strength, (ii) strong competition - interspecies competition is significantly larger than intraspecies competition, and (iii) slow competitors - the species comprising one alliance are significantly less mobile than the species comprising the other alliance. We employ asymptotic and perturbation methods to determine the outcome of the invasion (winning alliance) and compare our analytic results (generally very positively) with both numerical computations and, when parameters are such that the problem can be reduced to the two-species problem, with previously obtained results.

We consider cyclic ecological systems consisting of Nspecies u(i), i = 1,..., N, where N = 3 and N = 4. We employ a model in which species u(i) competes only with members of its own species (intraspecies competition), with a strength normalized to unity, and with species u(i+1), where i + 1 is taken to mean (imodN) + 1 (interspecies competition). The strength of the competition is quantified by an interaction coefficient alpha(i) > 0. When the interspecies competition is weak for all species (alpha(i) < 1 for all i), the only stable critical point is a unique coexistence state where all species coexist at nonzero concentrations (one might think that all species in the community form an alliance of the whole). All states involving extinction of at least one species are unstable. For strong competition (alpha(i) > 1 for all i), coexistence can still be stable under certain circumstances, but the predominant feature is complex dynamics for N = 3, involving heteroclinic cycles, spatiotemporal chaos, traveling waves, and modulated traveling waves, and interactions among stable alliances of noncompeting species for N = 4. All states other than the coexistence state involving alliances of competing species will be unphysical. We consider the case where one species (say, species j) is either a weak competitor in a community of strong competitors (alpha(j) > 1, alpha(i) < 1 for i not equal j), or a strong competitor in a community of weak competitors (alpha(j) < 1, alpha(i) > 1 for i not equal j). We show that in these cases, physical partial alliances of competing states are stable and there is a parameter regime where they represent the only stable states of the system. In all cases, we find that the behavior is dominated by transcritical bifurcations between two states: (i) the coexistence state and (ii) partial alliance states involving competing species. These transcritical bifurcations unify the behavior of three- and four-species systems for weak and strong exceptional species. We consider both homogeneous mixtures (i.e., ODEmodels) and the displacement of one state by another in both one and two dimensions (spatial dependence - systems of partial differential equations - PDEs). In the latter case, we develop and compare displacement speeds with approximate analytical results and show good agreement between computed and analytic speeds. (c) 2022 Elsevier Inc. All rights reserved.

Recently, it has been proposed to develop space power systems based on filtration combustion of metal powders with oxygen supplied by a chemical oxygen generator. The experiments with lithium and magnesium powders at natural infiltration of oxygen have shown propagation of both counterflow and coflow combustion waves. However, natural filtration combustion of metal powders at relatively low pressures is not sufficiently understood. In the present paper, we investigate the natural coflow combustion waves propagating through a porous medium. The porous matrix is made of metal particles that react with oxygen flowing from the open end of the sample to the reaction zone where it is consumed forming a condensed product which also has a porous structure. The gas flow is due to the pressure difference between the pressure at the open end and that in the reaction zone (the so-called natural filtration). The open end is where the sample is ignited, so that the gas flows through the reaction products, i.e. in the same direction as the combustion wave propagates (coflow filtration). Our mathematical model involves the conservation of energy equation and gas mass, solid reactant mass, and gas momentum balances, as well as an equation of state, and appropriate boundary and initial conditions. It is studied analytically under the combustion front approximation. When the reaction zone is close to the open end, there is sufficient amount of oxygen in the reaction zone and the reaction is controlled by kinetic factors (the kinetic regime of propagation). As the reaction moves away from the open end, it is gas supply that becomes a limiting factor (filtration regime). Both kinetic and filtration regimes of propagation as well the transition between them are analytically studied in this paper.

We consider a deterministic model of a rock-paper-scissors (RPS) cyclic ecological system. The model accounts for species mobility via Fickian diffusion, as well as interspecies interactions describing the cyclic competition scheme. The system admits both single-species equilibria (saddles) and a three -species coexistence equilibrium state. We focus on the regime where coexistence is unstable. When there is no spatial dependence, the solution of the corresponding ordinary differential equation (ODE) system admits an attracting heteroclinic cycle consisting of orbits connecting the three single-species saddles in the RPS sequence, to which most solutions with physical initial conditions asymptote. When there is spatial dependence (partial differential equation system), we consider mixed initial conditions (all species cohabit the same spatial region), as well as patch initial conditions, where each species is isolated from the other two, and show that, when interspecies competition is not strong, spatiotemporal chaotic behavior generally occurs. We propose a mechanism for the development of chaos---patch splitting-whereby sufficiently large patches are repeatedly split by the diffusion of competitors into the patch. However, when there is strong interspecies competition, ordered spatiotemporal patterns can occur. We consider both 1D patterns, corresponding to a community confined to a thin, circular annulus, and 2D geometries, simulating a petri dish. We show that, in 1 dimension, traveling arrays of single-species patches, as well as modulated traveling waves, consisting of patches which periodically expand and contract (breather modes), exist. In 2 dimensions, spirals, as well as localized patches chasing each other in the RPS sequence, can occur. We consider length and temporal scales appropriate for a bacterial community in a laboratory setting, qualitatively modeling observed Escherichia coli cyclic bacterial systems.

Combustion of a porous solid fuel is considered. An exothermic reaction takes place between the fuel and a gaseous oxidiser which is delivered to the reaction zone by filtration through the pores in the sample from an open end toward which the combustion wave propagates (counterflow filtration). The gas reacts with the solid fuel to form a solid product. The gas filtration is due to the pressure difference between the ambient pressure at the open end and the pressure in the reaction zone where the gas is being consumed (referred to as natural filtration). A 1D mathematical model based on equations describing conservation of energy, gas mass, solid reactant mass, and gas momentum, as well as an equation of state, and appropriate boundary and initial conditions is formulated and analytically studied taking advantage of the separation of length scales in the process. When the reaction zone is sufficiently far from the open end, the combustion wave propagates at a constant speed and has a time-independent structure, while when the reaction is close to the open end (closer than the filtration length), the structure of the combustion wave and its speed become time dependent. Both cases are discussed in the paper though the main emphasis is on short samples, in which the combustion wave is affected by the gas flow from the open end during the entire propagation process. A specific example of interest involves magnesium as the solid fuel and oxygen as the gaseous oxidiser.

We study traveling wave solutions of the Lotka-Volterra competition-diffusion system for two species with bistable nonlinearity. We analyze the speed of propagation and, most importantly, the sign of the speed, which determines the winner of the competition, in several limiting cases. One we refer to as the strong competition. Here the inter-species competition for each species is much stronger than the intra-species competition. Another is a moderate competition, where the inter-and intra-species competitions are close to each other in strength. We also analyze the cases of fast and slow competitors, i.e., when the diffusion coefficient of one of the species is much larger (or much smaller) than the other. We employ asymptotic and perturbation methods and support our approximate re-sults by direct numerical integration.(c) 2022 Elsevier Inc. All rights reserved.

We consider cyclic ecological systems consisting of more than three species, extending the well-studied three-species rock–paper–scissors (RPS) system. For such systems alliance and partial alliance states consisting of non-competing species can form, a phenomenon not present in classical RPS systems. We develop, analyze and simulate a May–Leonard type model accounting for cyclic dynamics and spatial diffusion. We analyze in detail four-species and five-species systems, determine dynamics for both spatially homogeneous initial conditions (i.e., ode models) and the propagation and interaction of patches of different species driven by diffusion and the cyclic dynamics (pde models). We show markedly different behavior between the four-and five-species models, with alliance states as stable steady states for the former whereas both coexistence of all species and heteroclinic cycles are possible stable outcomes for the latter. Finally, we discuss the extension of this even/odd split to systems with more than five species.

A rock–paper–scissors three species cyclic ecosystem is considered. Deterministic mathematical models based on delayed ODEs and nonlocal PDEs are proposed and studied both analytically and numerically. Transitions between the coexistence state that is associated with biodiversity, limit cycles and the heteroclinic cycle are discussed for the ODE model. Traveling waves between the coexistence state and single species states are studied for the PDE model. We show that delay promotes oscillatory instabilities of the coexistence state while nonlocality promotes stationary cellular instabilities.

In this paper, we consider a three-species food chain model with ratio-dependent predation, where species u is preyed upon by species v, which in turn is preyed upon by species w. Our primary focus is on biological control, where the bottom species u is an important crop, and v is a pest that has infested the crop. The superpredator w is introduced into this pest-infested environment in an attempt to restore the system to a pest-free state. We assume that the species can behave nonlocally, where individuals will interact over a distance, and incorporate this nonlocality into the model. For this model, we consider two types of nonlocality: one where the crop species u competes nonlocally with itself, and the other where the superpredator w is assumed to be highly mobile and therefore preys upon the pest v in a nonlocal fashion. We examine how biological control can prove to be highly susceptible to noise, and can fail outright if the pest species is highly diffusive. We show, however, that control can be restored if the superpredator is sufficiently diffusive, and that robust partial control can occur if the superpredator behaves nonlocally. Since the superpredator is generally introduced artificially, our results point to properties of the superpredator which can lead to successful control.

We consider predator invasions for the nonlocal Holling–Tanner model. Predators are introduced in a small region adjacent to an extensive predator-free region. In its simplest form an invasion front propagates into the predator-free region with a predator–prey coexistence state displacing the predator-free state. However, patterns may form in the wake of the invasion front due to instability of the coexistence state. The coexistence state can be subject to either oscillatory or cellular instability, depending on parameters. Furthermore, the oscillatory instability can be either at zero wave number or finite wave number. In addition, the (unstable) predator-free state can be subject to additional cellular instabilities when the extent of the nonlocality is sufficiently large. We perform numerical simulations that demonstrate that the invasion wave may have a complex structure in which different spatial regions exhibit qualitatively different behaviors. These regions are separated by relatively narrow transition regions that we refer to as fronts. We also derive analytic approximations for the speeds of the fronts and find qualitative and quantitative agreement with the results of computations.

We consider an analytically tractable switching model that is a simplification of a nonlocal, nonlinear reaction-diffusion model of population growth where we take the source term to be piecewise linear. The form of this source term allows us to consider both the monostable and bistable versions of the problem. By transforming to a traveling frame and choosing specific kernel functions, we are able to reduce the problem to a system of algebraic equations. We construct solutions and examine the propagation speed and monotonicity of the resulting waves.

We consider a model of two competing species with nonlocal competition for resources. The net birthrate is cubic, so that the model allows simulation of the Allee effect, whereby extinction is stable and intermediate populations promote growth, while saturation occurs via cubic competition terms. The model includes both interspecies and intraspecies nonlocal competition which enters via convolution integrals with a specified asymmetric competition kernel function. We introduce two parameters, δ, describing the extent of the coupling, with δ = 0 corresponding to local coupling, and α, describing the extent of the asymmetry, with α = 0 corresponding to symmetric nonlocal interactions. We consider the case where the local model admits a stable coexistence (populations of both species positive) equilibrium solution. We perform a linear stability analysis and show that this solution can be destabilized by sufficient nonlocality, i.e., when δ increases beyond a critical value. We then consider nonlinear patterns, far from the stability boundary. We show that nonlinear patterns consist of arrays of islands, regions of nonzero population, separated by deadzones, where the populations are essentially extinct, (with the array propagating in the case α ≠ 0). The predominant effect of the cubic model is that the islands for the two species are disjoint, so that each species lives in the deadzone of the other species. In addition, some patterns involve both hospitable and inhospitable deadzones, so that islands form in only some of the deadzones.

We study a reaction-diffusion population model for two competing species, u and v, which also includes nonlocal integral terms that represent competition for resources. The integral terms have the form of a convolution of a given kernel function and the solution. They manifest the fact that consumption of resources by the species at a spatial location x depends not just on the populations at point x but rather on weighted averages of the populations in an interval about x. The kernel functions that we employ are characterized by two parameters, delta, which gives a spatial scale of the nonlocality, with delta = 0 corresponding to the local case, and alpha, a parameter associated with the extent of the asymmetry of the kernel, where alpha = 0 corresponds to a symmetric kernel, i.e., to a kernel that is an even function of its argument.We consider the parameter regime where for delta = 0 the system admits a stable coexistence equilibrium, which is destabilized for sufficiently large delta. Deeply in the instability region, the ensuing patterns involve arrays of islands, regions of nonzero population, separated by deadzones where the populations are essentially extinct. These structures are stationary if alpha = 0 and propagate if alpha not equal 0. Unlike previous work, we study the patterns when the kernel parameters are different for the two species, focusing first on different. delta with alpha = 0 and then on different alpha with the same delta.In the first case we numerically find stationary patterns consisting of (i) plateau regions for the more local species (v), where the plateau value is essentially the v-component of the unstable equilibrium point, (ii) oscillatory patterns, where oscillations appear along the plateau and along the u-island and (iii) split patterns, where instead of an island, there is an island cluster of two or more neighbors separated by deadzones or near deadzones.In the second case we consider weakly coupled systems and show that there are two kinds of patterns, (i) bound waves, when each species propagates with the same speed and (ii) unbound patterns where each species propagates with its own speed, interacting with the other species when they intersect, i.e., when the two species occupy the same or closely spaced regions. We show that whether the patterns are bound or unbound depends on the stability of the decoupled system, i.e., the system when the coupling parameters are zero. When one of the species for the decoupled system is stable, then the ensuing patterns for weak coupling is bound, while if both species are unstable for the uncoupled systems, the resulting patterns are unbound for weak coupling.

We consider the use of step-functions to model Arrhenius reaction terms for traveling wave solutions to combustion problems. We develop a methodology by which the Arrhenius reaction rate term is replaced by a suitably normalized step-function. The resulting model introduces interior interfaces and allows the conservation equations for energy and species to be solved explicitly within the subdomains bounded by the interfaces. The problem can then be reduced to a small number of nonlinear algebraic equations governing appropriate interface conditions. We apply this methodology to a variety of single-reaction problems and show that the resulting solutions agree with those obtained by the well-known front δ-function) approximations for large Zeldovich numbers. We then consider multiple reaction problems, specifically problems involving two independent reactions and problems involving sequential reactions. For these problems we compare the results with simpler front models as well as with Arrhenius kinetics. We show that the step-function models are generally superior to the front models where available and agree, both qualitatively and with reasonable quantitative accuracy, with solutions obtained via full Arrhenius kinetics.

We consider a model of two competing species with asymmetric nonlocal coupling in a competition for resources. The nonlocal coupling is via convolution integrals and the asymmetry is via convolution kernel functions which are not even functions of their arguments. The nonlocality is due to species mobility, so that at any fixed point in space the competition for resources depends not just on the populations at that point but on a suitably weighted average of the populations. We introduce two parameters, δ, describing the extent of the coupling, with δ=0 corresponding to local coupling, and α, describing the extent of the asymmetry, with α=0 corresponding to symmetric nonlocal interactions. We consider the case where the model admits a stable coexistence equilibrium solution. We perform a linear stability analysis and show that this solution can be destabilized by sufficient nonlocality, i.e., when δ increases beyond a critical value. We consider two specific kernel functions, (i) an asymmetric Gaussian and (ii) an asymmetric stepfunction. We compute the stability boundary as a function of α, and for δ beyond the stability boundary we determine unstable wavenumber bands. We compute nonlinear patterns for δ significantly beyond the stability boundary. Patterns consist of arrays of islands, regions of nonzero population, separated by either near-deadzones where the populations are small, but nonzero, or by deadzones where populations are exponentially small and essentially extinct. We find solutions consisting of propagating traveling waves of islands, solutions exhibiting colony formation, where a colony is formed just ahead of an island and eventually grows as the parent island decays, and modulated traveling waves, where competition between the two species allows propagation and inhibits colony formation. We explain colony formation and the modulated traveling waves as due to a positive feedback mechanism associated with small variations in the amplitude of the parent island.

We analyze and compute an extension of a previously developed population model based on the well-known diffusive logistic equation with nonlocal interaction, to a system involving competing species. Our model involves a system of nonlinear integro-differential equations, with the nonlocal interaction characterized by convolution integrals of the population densities against specified kernel functions. The extent of the nonlocal coupling is characterized by a parameter δ so that when δ→0 the problem becomes local. We consider critical points of the model, i.e., spatially homogeneous equilibrium solutions. There is generally one critical point in the first quadrant (i.e., both population densities positive), denoting coexistence of the two species. We show that this solution can be destabilized by the nonlocal coupling and obtain general conditions for stability of this critical point as a function of δ, the specific kernel function and parameters of the model. We study the nonlinear behavior of the model and show that the populations can evolve to localized cells, or islands. We find that the stability transition is supercritical. Near the stability boundary solutions are small amplitude, nearly sinusoidal oscillations, however, when δ increases large amplitude, nonlinear states are found. We find a multiplicity of stable, steady state patterns. We further show that with a stepfunction kernel function the structure of these islands, a highly nonlinear phenomenon, can be described analytically. Finally, we analyze the role of the kernel function and show that for some choices of kernel function the resulting population islands can exhibit tip-splitting behavior and island amplitude modulation.

Spatiotemporal Turing-Hopf pinning solutions near the codimension-two Turing-Hopf point of the one-dimensional Brusselator model are studied. Both the Turing and Hopf bifurcations are supercritical and stable. The pinning solutions exhibit coexistence of stationary stripes of near critical wavelength and time-periodic oscillations near the characteristic Hopf frequency. Such solutions of this nonvariational problem are in contrast to the stationary pinning solutions found in the subcritical Turing regime for the variational Swift-Hohenberg equations, characterized by a spatially periodic pattern embedded in a spatially homogeneous background state. Numerical continuation was used to solve periodic boundary value problems in time for the Fourier amplitudes of the spatiotemporal Turing-Hopf pinning solutions. The solution branches are organized in a series of saddle-node bifurcations similar to the known snaking structures of stationary pinning solutions. We find two intertwined pairs of such branches, one with a defect in the middle of the striped region, and one without. Solutions on one branch of one pair differ from those on the other branch by a π phase shift in the spatially periodic region, i.e., locations of local minima of solutions on one branch correspond to locations of maxima of solutions on the other branch. These branches are connected to branches exhibiting collapsed snaking behavior, where the snaking region collapses to almost a single value in the bifurcation parameter. Solutions along various parts of the branches are described in detail. Time dependent depinning dynamics outside the saddle nodes are illustrated, and a time scale for the depinning transitions is numerically established. Wavelength variation within the snaking region is discussed, and reasons for the variation are given in the context of amplitude equations. Finally, we compare the pinning region to the Maxwell line found numerically by time evolving the amplitude equations.

We consider the use of step functions to model Arrhenius reaction terms for traveling wave solutions to combustion problems involving condensed-phase competing reactions such as those occurring in combustion synthesis via the self-propagating high-temperature synthesis process. For each reaction, the Arrhenius temperature dependence of the reaction rate is replaced by a step function. The resulting model introduces interior interfaces and allows the transport equations for energy and species to be solved explicitly within the subdomains bounded by these interfaces. The problem can then be reduced to a small number of algebraic equations describing appropriate interface conditions. We apply this methodology to study steady traveling waves with competing reactions, finding regions where multiple solutions are possible, and comparing our results with numerical calculations employing Arrhenius kinetics. We then analyze and determine regions of stability and instability for the traveling wave solutions.