Nonconstant Solutions Research Articles

Spatio-Temporal Patterns of Predator–Prey Model with Allee Effect and Constant Stocking Rate for Predator

A diffusive predator–prey model with Allee effect and constant stocking rate for predator is investigated and it is shown that Allee effect is the decisive factor driving the formation of Turing pattern. Furthermore, it is observed that Turing pattern appears only when the diffusion rate of the prey is faster than that of the predator, which is just opposite to the condition of Turing pattern in the classical predator–prey system. Some sufficient conditions are obtained to ensure the asymptotical stability of a spatially homogeneous steady-state solution. The existence and nonexistence of positive nonconstant steady-state solutions are investigated to understand the mechanisms of generating spatiotemporal patterns. Furthermore, Hopf and steady-state bifurcations are analyzed in detail by using Lyapunov–Schmidt reduction.

Three positive nonconstant radial solutions of nonlinear Neumann problems with indefinite weight

In this article, we consider the global behavior of components of positive nonconstant radial solutions for the Neumann problem with indefinite weight where is a parameter, is the unit open ball, and satisfying is the sign-changing function. We determine the intervals of λ in which the above problem has one, two or three positive nonconstant radial solutions by using the directions of a bifurcation.

Nondegenerate solutions for constrained semilinear elliptic problems on Riemannian manifolds

Let $$M^n$$ be a connected compact smooth manifold, where $$n \ge 2$$ . In this article, we prove that nondegeneracy of nonconstant solutions for a class of singularly perturbed semilinear elliptic problems on M is generic with respect to the pair $$(\epsilon ,g)$$ , where $$\epsilon >0$$ and g is a metric of class $$C^k$$ , $$k\ge 1$$ . As applications, we show that under certain growth conditions, such result generalizes to nondegeneracy of any solution for the Allen-Cahn or nonlinear Schrödinger equations.

Effect of the protection zone in a diffusive ratio-dependent predator\u2013prey model with fear and Allee effect

In this paper, we study the influence of a protection zone for the prey on a diffusive predator–prey model with fear factor and Allee effect. The prior estimate, global existence, nonexistence of nonconstant positive solutions and bifurcation from semitrivial solutions are well discussed. We show the existence of a critical patch value lambda ^{D}_{1}(Omega _{0}) of the protection zone, described by the principal eigenvalue of the Laplacian operator over Omega _{0} with Neumann boundary conditions. When the mortality rate of the predator mu geq d_{2}lambda ^{D}_{1}(Omega _{0}), we show that the semitrivial solutions (1,0) and (theta,0) are unstable and there is no bifurcation occurring along respective semitrivial branches.

Pattern formation of a biomass–water reaction–diffusion model

In this paper, we are concerned with a biomass–water reaction–diffusion model subject to the homogeneous Neumann boundary condition. We derive several sufficient conditions on the existence and non-existence of non-constant stationary solutions with respect to large or small diffusion rate, which give the criteria for the possibility of Turing patterns in this system. Our results confirm the numerical findings of Manor and Shnerb, (2006) and also complement the theoretical results of Wang et al., (2017) for the corresponding ODE model.

Spatiotemporal dynamic analysis of an extended water-plant model with power exponent plant growth and nonlocal plant loss

In this paper, we offer a detailed mathematical analysis of an extended water-plant model to describe vegetation pattern formation in arid and semi-arid grazing ecosystems. We present some fundamental analytic properties of the ordinary differential equations and the nonlocal reaction-diffusion equations. The results indicate that when infiltration parameter is equal to one, the unique positive equilibrium is locally stable and where diffusion cannot lead to Turing instability. On the contrary, when infiltration parameter is greater than one, the existence of both positive equilibriums and Turing instability can be observed under further parametric constraints. In addition, we report some characterizations for the non-constant positive steady state solutions, including a priori estimate of the positive solutions and the non-existence and existence of non-constant positive solutions. Some notes on numerical simulation are given, based on different diffusion coefficients, infiltration parameter and rain fall parameter.

Infinitely many sign-changing solutions of a critical fractional equation

In this paper, we obtain results of nonexistence of nonconstant positive solutions, and also existence of an unbounded sequence of sign-changing solutions for some critical problems involving conformally invariant operators on the unit sphere, in particular to the fractional Laplacian operator in the Euclidean space. Our arguments are based on a reduction of the initial problem in the Euclidean space to an equivalent problem on the standard unit sphere and vice versa, what together with blow up arguments, a variant of Pohozaev’s type identity, a refinement of regularity results for this type operators, and finally, by exploiting the symmetries of the sphere.

Minimal wave speed in an HIV-1 virus integrodifference system

This paper is concerned with the minimal wave speed of nonconstant traveling wave solutions in an HIV-1 virus integrodifference system. Here, the traveling wave solution models the spatial spreading process of infected cells and virus. When the basic reproduction ratio of the corresponding ordinary differential system or difference system is larger than one, we establish the existence of nonconstant traveling wave solutions if the wave speed is not less than a threshold, and if the speed is smaller than the threshold, we prove the nonexistence of nonconstant traveling wave solutions. Moreover, when the basic reproduction ratio of the corresponding ordinary differential system or difference system is not larger than one, we also confirm the nonexistence of nonconstant traveling wave solutions.

A model of the burglar alarm hypothesis of prey alarm calls

When approached by a predator many prey species will emit an “alarm call” as a form of anti-predator behavior. One hypothesis for the function of alarm calls is the “burglar alarm” hypothesis whereby upon attack, a prey renders itself dangerous to a predator by generating an alarm call that attracts a predator at higher trophic levels in the food chain; that is, attracts a predator to the prey’s own predator. This paper concerns a model incorporating a mechanism to test the burglar alarm hypothesis. We prove in one space dimension global existence, of positive bounded classical solutions, and establish existence of non-constant equilibrium solutions and assess their stability. We provide some representative numerical simulations to emphasize the nature of pattern formation for this model and demonstrate the benefit achieved by a signal inducing prey species under the burglar alarm hypothesis.

Global stability of spatially nonhomogeneous steady state solution in a diffusive Holling-Tanner predator-prey model

The global stability of the nonhomogeneous positive steady state solution to a diffusive Holling-Tanner predator-prey model in a heterogeneous environment is proved by using a newly constructed Lyapunov function and estimates of nonconstant steady state solutions. The techniques developed here can be adapted for other spatially heterogeneous consumer-resource models.

Steady states and pattern formation of the density-suppressed motility model

Abstract This paper considers the stationary problem of density-suppressed motility models proposed in Fu et al. (2012) and Liu et al. (2011) in one dimension with Neumman boundary conditions. The models consist of parabolic equations with cross-diffusion and degeneracy. We employ the global bifurcation theory and Helly compactness theorem to explore the conditions under which non-constant stationary (pattern) solutions exist and asymptotic profiles of solutions as some parameter value is small. When the cell growth is not considered, we are able to show the monotonicity of solutions and hence achieve a global bifurcation diagram by treating the chemical diffusion rate as a bifurcation parameter. Furthermore, we show that the solutions have boundary spikes as the chemical diffusion rate tends to zero and identify the conditions for the non-existence of non-constant solutions. When transformed to specific motility functions, our results indeed give sharp conditions on the existence of non-constant stationary solutions. While with the cell growth, the structure of global bifurcation diagram is much more complicated and in particular the solution loses the monotonicity property. By treating the growth rate as a bifurcation parameter, we identify a minimum range of growth rate in which non-constant stationary solutions are warranted, while a global bifurcation diagram can still be attained in a special situation. We use numerical simulations to test our analytical results and illustrate that patterns can be very intricate and stable stationary solutions may not exist when the parameter value is outside the minimal range identified in our paper.

Steady states of Lotka-Volterra competition models with nonlinear cross-diffusion

This paper is concerned with a Lotka-Volterra model with nonlinear cross diffusion. The global existence of generalized solutions is established under some proper assumptions, and the nonexistence of nonconstant solutions is also investigated when diffusion rate is sufficiently large. At the same time, sufficient conditions ensuring the existence of non-constant solutions are obtained by using Leray-Schauder degree theory. Furthermore, steady-state bifurcation analysis is carried out in details by using Lyapunov-Schmidt reduction.

Patterns with prescribed numbers of critical points on topological tori

We study the existence of critical points of stable stationary solutions to reaction–diffusion problems on topological tori. Stable nonconstant stationary solutions are often called patterns. We construct topological tori and patterns with prescribed numbers of critical points whose locations are explicit.

The Economic Infeasibility of Salinity Gradient Energy via Pressure Retarded Osmosis

Pressure retarded osmosis (PRO) harvests the chemical potential difference between water sources of two different salinities and has been widely researched as a source of low carbon energy. This study identifies the cost-optimal design and operation using a nonlinear programming model to minimize the levelized cost of electricity (LCOE) of a PRO system for specified process and financial parameters. Our model includes a detailed finite difference module-scale model that considers the pressure drop along the module, nonconstant solution properties, local mass transfer coefficients, and equipment inefficiencies that previous analyses lacked. Under realistic case specifications and present-day costs, the expected median LCOE ranges from $2.37/kWh (seawater draw) to $0.11/kWh (Dead Sea draw). The optimistic case specifications, which assume significant improvements in process performance with no additional increase in cost, lower these median costs to $1.00/kWh (seawater draw) and $0.05/kWh (Dead Sea draw). These estimates exclude variable pretreatment costs and are likely to underestimate the actual LCOE. Membrane costs and maintenance are the primary cost components for all cases. Comparing the LCOE and capacity factor of PRO to other low carbon energy sources suggests that reductions in PRO component costs are unlikely to make PRO cost competitive with renewable energy technologies.

Conley index for manifold-valued retarded functional differential equations without uniqueness of solutions

We develop a Conley index theory for retarded functional differential equations $\dot x=f(x_{t})$ with values in a differentiable manifold and (merely) continuous nonlinearities f. We use this index to establish an existence result for nonconstant full solutions of such equations.

Travelling wave solutions in a predator–prey integrodifference system

This paper studies the minimal wave speed of travelling wave solutions in a predator–prey integrodifference system. Even if the predator vanishes, the corresponding scalar equation of prey may be nonmonotonic. Without the assumption of classical comparison principle in this coupled system, we investigate travelling wave solutions modelling the process that the predator invades the habitat of the prey. By showing the existence and nonexistence of nonconstant travelling wave solutions, the minimal wave speed of travelling wave solutions is confirmed. To achieve the purpose, we utilize recipes of constructing generalized upper and lower solutions, introducing auxiliary equations and applying the propagation theory of scalar equations.

Nonconstant positive solutions to the ratio-dependent predator-prey system with prey-taxis in one dimension

<p style='text-indent:20px;'>Resorting to M.G. Crandall and P.H. Rabinowitz's well-known bifurcation theory we first obtain the local structure of steady states concerning the ratio–dependent predator–prey system with prey-taxis in spatial one dimension, which bifurcate from the homogeneous coexistence steady states when treating the prey–tactic coefficient as a bifurcation parameter. Based on this, then the global structure of positive solution is established. Moreover, through asymptotic analysis and eigenvalue perturbation we find the stability criterion of such bifurcating steady states. Finally, several numerical simulations are performed to show the pattern formation.</p>

Spatiotemporal patterns of a diffusive plant–herbivore model with toxin-determined functional responses: Multiple bifurcations

In this paper, a homogeneous diffusive plant–herbivore model with toxin-determined functional response subject to the homogeneous Neumann boundary condition in the one dimensional spatial open bounded domain is considered. By using Hopf bifurcation theorem and steady state bifurcation theorem due to Yi et al. (2009), we are able to show the existence of Hopf bifurcating periodic solutions (spatially homogeneous and non-homogeneous) and bifurcating non-constant steady state solutions. In particular, under certain conditions, the globally asymptotic stability of the positive constant steady state solutions and the non-existence of non-constant positive steady state solutions are investigated. These results allow the clear understanding of the mechanisms of the spatiotemporal pattern formations of this ecology model. In particular, our numerical results authenticate that toxicant parameter will play important roles in the stability and instability of the periodic solutions.

Analysis and simulation on dynamics of a partial differential system with nonlinear functional responses

We introduce a reaction–diffusion system with modified nonlinear functional responses. We first discuss the large-time behavior of positive solutions for the system. And then, for the corresponding steady-state system, we are concerned with the priori estimate, the existence of the nonconstant positive solutions as well as the bifurcations emitting from the positive constant equilibrium solution. Finally, we present some numerical examples to test the theoretical and computational analysis results. Meanwhile, we depict the trajectory graphs and spatiotemporal patterns to simulate the dynamics for the system. The numerical computations and simulated graphs imply that the available food resource for consumer is very likely not single.

The Exact Entire Solutions of Certain Type of Nonlinear Difference Equations

In this paper, we consider the entire solutions of nonlinear difference equation $$f^{3}+q(z)\Delta f=p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z}$$ , where $$q$$ is a polynomial, and $$p_{1},p_{2},\alpha_{1},$$ and $$\alpha_{2}$$ are nonzero constants with $$\alpha_{1}\neq\alpha_{2}$$ . It is showed that if $$f$$ is a nonconstant entire solution of $$\rho_{2}(f)<1$$ to the above equation, then $$f(z)=e_{1}e^{\frac{\alpha_{1}z}{3}}+e_{2}e^{\frac{\alpha_{2}z}{3}}$$ , where $$e_{1}$$ and $$e_{2}$$ are two constants. Meanwhile, we give an affirmative answer to the conjecture posed by Zhang et al in [18].