# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

April 2021 , Volume 26 , Issue 4

Special issue for Prof. Sze-Bi Hsu

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2021, 26(4): ⅰ-ⅱ doi: 10.3934/dcdsb.2021071 +[Abstract](263) +[HTML](95) +[PDF](83.14KB)
Abstract:
2021, 26(4): 1763-1781 doi: 10.3934/dcdsb.2021005 +[Abstract](326) +[HTML](97) +[PDF](566.79KB)
Abstract:

For a stage-structured population model in periodic discrete habitat, with periodic initial values it reduces to a system of two differential equations with time delay. Assuming the birth rate is of unimodal type, we obtain the influence of time delay on the local and global dynamics. In particular, large delay leads to population vanishing. As delay decreases, we found three critical values of delay for the emergence of different dynamics, by appealing to a combination of monotone dynamical system theory, Hopf bifurcation theory and the fluctuation method. Numerical simulations are also performed to illustrate the results.

2021, 26(4): 1783-1795 doi: 10.3934/dcdsb.2020361 +[Abstract](272) +[HTML](102) +[PDF](331.13KB)
Abstract:

We analyze a reaction-diffusion system modeling the competition of multiple phytoplankton species which are limited only by light. While the dynamics of a single species has been well studied, the dynamics of the two-species model has only begun to be understood with the recent establishment of a comparison principle. In this paper, we show that the competition of \begin{document}$N$\end{document} similar phytoplankton species, for any number \begin{document}$N$\end{document}, generically leads to competitive exclusion. The main tool is the theory of a normalized principal bundle for linear parabolic equations.

2021, 26(4): 1797-1809 doi: 10.3934/dcdsb.2021028 +[Abstract](239) +[HTML](84) +[PDF](341.04KB)
Abstract:

We study a free boundary problem of two competing species with latent heat effect. We establish the existence and uniqueness of the traveling wave solution and derive upper and lower bounds for the wave speed. Especially our results show that the latent heat retards propagation of the waves.

2021, 26(4): 1811-1825 doi: 10.3934/dcdsb.2020040 +[Abstract](1223) +[HTML](479) +[PDF](345.36KB)
Abstract:

This work is concerned with the dynamical behaviors of a singular predator-prey model. We first review some well-known results obtained recently. Then we give some new results on the spreading speed of the predator, the existence vs non-existence of traveling waves connecting the predator-free state to the co-existence state, and the existence vs non-existence of spatially periodic traveling waves to this singular predator-prey system.

2021, 26(4): 1827-1842 doi: 10.3934/dcdsb.2020364 +[Abstract](261) +[HTML](108) +[PDF](341.26KB)
Abstract:

The quadratic autocatalytic reaction forms a key step in a number of chemical reaction systems, and traveling waves are observed in such systems. In this study, we investigate the effect of complexation reactions on traveling waves in the quadratic autocatalytic reaction system. More precisely, under the assumption that the complexation reaction is fast relative to the autocatalytic reaction, we show that the governing system is reduced to a two-component reaction-diffusion system with density-dependent diffusivity. Further, the numerical evidence suggests that for some parameter values, a traveling wave solution of this reduced two-component system is nonlinearly selected. This is contrast to that associated with the quadratic autocatalytic reaction (without complexation reactions).

2021, 26(4): 1843-1866 doi: 10.3934/dcdsb.2020042 +[Abstract](1212) +[HTML](582) +[PDF](725.47KB)
Abstract:

The stability of a constant steady state in a general reaction-diffusion activator-inhibitor model with nonlocal dispersal of the activator or inhibitor is considered. It is shown that Turing type instability and associated spatial patterns can be induced by fast nonlocal inhibitor dispersal and slow activator diffusion, and slow nonlocal activator dispersal also causes instability but may not produce stable spatial patterns. The existence of nonconstant positive steady states is shown through bifurcation theory. This suggests a new mechanism for spatial pattern formation, which has different instability parameter regime compared to Turing mechanism. The theoretical results are applied to pattern formation problems in nonlocal Klausmeier-Gray-Scott water-plant model and Holling-Tanner predator-prey model.

2021, 26(4): 1867-1887 doi: 10.3934/dcdsb.2020363 +[Abstract](228) +[HTML](101) +[PDF](590.45KB)
Abstract:

Experiments showed that a neuron can fire when its membrane potential (an intrinsic quality related to its membrane electrical charge) reaches a specific threshold. On theoretical studies, there are two crucial issues in exploring cortical neuronal dynamics: (i) what model describes spiking dynamics of each neuron, and (ii) how the neurons are connected [E. M. Izhikevich, IEEE Trans. Neural Networks, 15 (2004)]. To study the first issue, we propose the time delay effect on the well-known integrate-and-fire (IF) model which is classically introduced to study the spiking behaviors in neural systems by using the spike-and-reset procedure. Under the consideration of delayed adaptation on the membrane potential, the parameter range for the IF model with spiking dynamics becomes wider due to undergoing subcritical Hopf bifurcation and the existence of an unstable orbit. To study the second issue, we consider the system with two coupled identical IF units where time delay takes place in the coupling structure. We also demonstrate spiking behaviors in the coupled system when the delay time is large enough, and it contributes an original viewpoint of the connection between neurons. In contrast with the emergence of delay-induced spiking in a single-neuron system, a coupled two-neuron system involve both emergence and death of spiking according to different values of delay times. We also discuss the ranges of different parameters in which it allows occurrence of spiking behaviors.

2021, 26(4): 1889-1915 doi: 10.3934/dcdsb.2020281 +[Abstract](310) +[HTML](176) +[PDF](1560.4KB)
Abstract:

In this paper, we formulate an ODE model to describe the population dynamics of one non-dispersing prey and two dispersing predators in a two-patch environment with spatial heterogeneity. The dispersals of the predators are implicitly reflected by the allocation of their presence (foraging time) in each patch. We analyze the dynamics of the model and discuss some biological implications of the theoretical results on the dynamics of the model. Particularly, we relate the results to the evolution of the allocation strategy and explore the impact of the spatial heterogeneity and the difference in fitness of the two predators on the allocation strategy. Under certain range of other parameters, we observe the existence of an evolutionarily stable strategy (ESS) while in some other ranges, the ESS disappears. We also discuss some possible extensions of the model. Particularly, when the model is modified to allow distinct preys in the two patches, we find that the heterogeneity in predation rates and biomass transfer rates in the two patches caused by such a modification may lead to otherwise impossible bi-stability for some pairs of equilibria.

2021, 26(4): 1917-1930 doi: 10.3934/dcdsb.2020082 +[Abstract](911) +[HTML](467) +[PDF](309.99KB)
Abstract:

The beginning of the transition from a hunter-gatherer way of life to a more settled, farming-based one in Europe is dated to the Neolithic period. The spread of farming culture from the Middle East is associated, among other things, with the transformation of landscape, cultivation of domesticated plants, domestication of animals, as well as it is identified with the distribution of certain human genetic lineages. Ecological models attribute the Neolithic transition either to the spread of the initial farming populations or to the dispersal of farming knowledge and ideas with the simultaneous conversion of hunter-gatherers to farmers. A reaction-diffusion model proposed by Aoki, Shida and Shigesada in 1996 is the first model that includes the populations of initial farmers and converted farmers from hunter-gatherers. Both populations compete for the same resources in this model, however, otherwise they evolve independently of each other from a genetic point of view. We study the large time behaviour of solutions to this model in bounded domains and we explain which farmers under what conditions dominate over the other and eventually occupy the whole habitat.

2021, 26(4): 1931-1966 doi: 10.3934/dcdsb.2020326 +[Abstract](383) +[HTML](204) +[PDF](506.05KB)
Abstract:

In this work we describe a hyperbolic model with cell-cell repulsion with a dynamics in the population of cells. More precisely, we consider a population of cells producing a field (which we call "pressure") which induces a motion of the cells following the opposite of the gradient. The field indicates the local density of population and we assume that cells try to avoid crowded areas and prefer locally empty spaces which are far away from the carrying capacity. We analyze the well-posed property of the associated Cauchy problem on the real line. Moreover we obtain a convergence result for bounded initial distributions which are positive and stay away from zero uniformly on the real line.

2021, 26(4): 1967-1990 doi: 10.3934/dcdsb.2020041 +[Abstract](1147) +[HTML](507) +[PDF](1178.03KB)
Abstract:

In this paper, we study the global dynamics of a density-dependent predator-prey system with ratio-dependent functional response. The main features and challenges are that the origin of this model is a degenerate equilibrium of higher order and there are multiple positive equilibria. Firstly, local qualitative behavior of the system around the origin is explicitly described. Then, based on the dynamics around the origin and other equilibria, global qualitative analysis of the model is carried out. Finally, the existence of Bogdanov-Takens bifurcation (cusp case) of codimension two is analyzed. This shows that the system undergoes various bifurcation phenomena, including saddle-node bifurcation, Hopf bifurcation, and homoclinic bifurcation along with different topological sectors near the degenerate origin. Numerical simulations are presented to illustrate the theoretical results.

2021, 26(4): 1991-2010 doi: 10.3934/dcdsb.2020362 +[Abstract](251) +[HTML](97) +[PDF](445.22KB)
Abstract:

Huang et al. [10] developed a hybrid continuous/discrete-time model to describe the persistence and invasion dynamics of Zebra mussels in rivers. They used a net reproductive rate \begin{document}$R_0$\end{document} to determine population persistence in a bounded domain and estimated spreading speeds by applying the linear determinacy conjecture and using the formula in [16]. Since the associated solution operator is non-monotonic and non-compact, it is nontrivial to rigorously establish these quantities. In this paper, we analyze the spatial dynamics of this model mathematically. We first solve the parabolic equation and rewrite the model into a fully discrete-time model. In a bounded domain, we show that the spectral radius \begin{document}$\hat{r}$\end{document} of the linearized operator can be used to determine population persistence and that the sign of \begin{document}$\hat{r}-1$\end{document} is the same as that of \begin{document}$R_0-1$\end{document}, which confirms that \begin{document}$R_0$\end{document} defined in [10] can be used to determine population persistence. In an unbounded domain, we construct two monotonic operators to control the model operator from above and from below and obtain upper and lower bounds of the spreading speeds of the model.

2021, 26(4): 2011-2024 doi: 10.3934/dcdsb.2020188 +[Abstract](552) +[HTML](259) +[PDF](339.74KB)
Abstract:

In this work, we are concerned with the problem of estimating the size of an inclusion embedded in an object laying in the two dimensional domain. We assume that the object is occupied by an exotic material which obeys a nonlinear Ohms' law. In view of the assumption of the power law, we thus consider the weighted \begin{document}$p$\end{document}-Laplace equation as a model problem in this case. Using only one voltage-current measurement, we give upper and lower bounds of the size of the inclusion.

2021, 26(4): 2025-2035 doi: 10.3934/dcdsb.2020358 +[Abstract](257) +[HTML](113) +[PDF](364.22KB)
Abstract:

In this paper, we study the continuous-time nearest stable matrix problem: given a \begin{document}$2\times 2$\end{document} real matrix \begin{document}$A$\end{document}, minimize the Frobenius norm of \begin{document}$A-X$\end{document}, where \begin{document}$X$\end{document} is a stable matrix. We provide an explicit formula for the global minimizer \begin{document}$X_*$\end{document}. The uniqueness of the minimizer is also studied.

2021, 26(4): 2037-2053 doi: 10.3934/dcdsb.2020365 +[Abstract](261) +[HTML](100) +[PDF](433.46KB)
Abstract:

In this paper, we consider the hermitian Riccati difference equations. Analogous to a Riccati differential equation, there is a connection between a Riccati difference equation and its associated linear difference equation. Based on the linear difference equation, we can obtain an explicit representation for the solution of the Riccati difference equation and define the extended solution. Further, we can characterize the asymptotic state of the extended solution and the rate of convergence. Constant equilibrium solutions, periodic solutions and closed limit cycles are exhibited in the investigation of asymptotic behavior of the hermitian Riccati difference equations.

2021, 26(4): 2055-2065 doi: 10.3934/dcdsb.2020280 +[Abstract](363) +[HTML](176) +[PDF](331.85KB)
Abstract:

This paper focuses on an optimization problem arising in population biology. We investigate the effect of the resources distribution and the migration rate on the total population size of some species, which migrates among patches with the identical probability and grows logistically in each patch. We aim to maximize the total population size by the distribution of resources and the rate of migration.

2021, 26(4): 2067-2084 doi: 10.3934/dcdsb.2020325 +[Abstract](288) +[HTML](144) +[PDF](430.2KB)
Abstract:

In the paper, we study a class of semilinear fractional semilinear elliptic equations involving concave-convex nonlinearities:

where \begin{document}$\alpha\in (0,1]$\end{document}, \begin{document}$1<q<2<p<2_{\alpha}^{\ast }\ \left( 2_{\alpha}^{\ast } = \frac{2N}{N-2\alpha}\text{ for}\ N> 2\alpha\right),$\end{document} the potential \begin{document}$V_{\lambda }(x) = \lambda a(x)-b(x)$\end{document} and the parameter \begin{document}$\lambda >0.$\end{document} Under some suitable assumptions on \begin{document}$a,b$\end{document} and the weight functions \begin{document}$f,g$\end{document}, we obtain the existence and multiplicity of non-trivial (positive) solutions for \begin{document}$\lambda$\end{document} large enough. An interesting phenomenon is that we do not need the condition that weight functions \begin{document}$f, g$\end{document} are integrable or bounded on whole space \begin{document}$\mathbb{R}^{N}.$\end{document}

2021, 26(4): 2085-2113 doi: 10.3934/dcdsb.2020081 +[Abstract](1523) +[HTML](1304) +[PDF](494.69KB)
Abstract:

This paper introduces a spatially heterogeneous diffusive predator–prey model unifying the classical Lotka–Volterra and Holling–Tanner ones through a prey saturation coefficient, \begin{document}$m(x)$\end{document}, which is spatially heterogenous and it is allowed to 'degenerate'. Thus, in some patches of the territory the species can interact according to a Lotka–Volterra kinetics, while in others the prey saturation effects play a significant role on the dynamics of the species. As we are working under general mixed boundary conditions of non-classical type, we must invoke to some very recent technical devices to get some of the main results of this paper.

2021, 26(4): 2115-2132 doi: 10.3934/dcdsb.2020359 +[Abstract](240) +[HTML](95) +[PDF](375.19KB)
Abstract:

This paper analytically investigates a nonlocal system of reaction-diffusion-advection equations modeling the competition of two phytoplankton species for a limiting nutrient and light in a water column, where dead phytoplankton species can get recycled back into the system as a resource for growth. The threshold dynamics of the single population model is first established. Then the utilization of abstract persistence theory enables us to show that two species population system admits a coexistence steady state and the system is uniformly persistent if the trivial steady state and two global attractors on the boundary are all weak repellers.

2021, 26(4): 2133-2159 doi: 10.3934/dcdsb.2020138 +[Abstract](892) +[HTML](428) +[PDF](1052.36KB)
Abstract:

Immune checkpoint inhibitors (ICIs) are a novel cancer therapy that may induce tumor regression across multiple types of cancer. There has recently been interest in combining the ICIs with other forms of treatments, as not all patients benefit from monotherapy. We propose a mathematical model consisting of ordinary differential equations to investigate the combination treatments of the ICI avelumab and the immunostimulant NHS-muIL12. We validated the model using the average tumor volume curves provided in Xu et al. (2017). We initially analyzed a simple generic model without the use of any drug, which provided us with mathematical conditions for local stability for both the tumorous and tumor-free equilibrium. This enabled us to adapt these conditions for special cases of the model. Additionally, we conducted systematic mathematical analysis for the case that both drugs are applied continuously. Numerical simulations suggest that the two drugs act synergistically, such that, compared to monotherapy, only about one-third the dose of both drugs is required in combination for tumor control.

2021, 26(4): 2161-2172 doi: 10.3934/dcdsb.2021014 +[Abstract](254) +[HTML](105) +[PDF](519.09KB)
Abstract:

We investigate the dynamics of the Poincar\begin{document}$\acute{\rm e}$\end{document}-map for an \begin{document}$n$\end{document}-dimensional Lotka-Volterra competitive model with seasonal succession. It is proved that there exists an \begin{document}$(n-1)$\end{document}-dimensional carrying simplex \begin{document}$\Sigma$\end{document} which attracts every nontrivial orbit in \begin{document}$\mathbb{R}^n_+$\end{document}. By using the theory of the carrying simplex, we simplify the approach for the complete classification of global dynamics for the two-dimensional Lotka-Volterra competitive model with seasonal succession proposed in [Hsu and Zhao, J. Math. Biology 64(2012), 109-130]. Our approach avoids the complicated estimates for the Floquet multipliers of the positive periodic solutions.

2021, 26(4): 2173-2199 doi: 10.3934/dcdsb.2021027 +[Abstract](179) +[HTML](70) +[PDF](4152.26KB)
Abstract:

In this paper we study the spruce-budworm interaction model with Holling's type II functional response. The existence, number and stability of equilibria are studied. Moreover, we prove the existence of relaxation oscillations by using singular perturbation method and give an asymptotic expression of the period of relaxation oscillations. Finally, the parameter ranges which allow the relaxation oscillations in several scenarios are explored and displayed by conducting numerical simulations.

2021, 26(4): 2201-2238 doi: 10.3934/dcdsb.2020360 +[Abstract](277) +[HTML](103) +[PDF](529.39KB)
Abstract:

In this paper, we determine the long-time dynamical behaviour of a reaction-diffusion system with free boundaries, which models the spreading of an epidemic whose moving front is represented by the free boundaries. The system reduces to the epidemic model of Capasso and Maddalena [5] when the boundary is fixed, and it reduces to the model of Ahn et al. [1] if diffusion of the infective host population is ignored. We prove a spreading-vanishing dichotomy and determine exactly when each of the alternatives occurs. If the reproduction number \begin{document}$R_0$\end{document} obtained from the corresponding ODE model is no larger than 1, then the epidemic modelled here will vanish, while if \begin{document}$R_0>1$\end{document}, then the epidemic may vanish or spread depending on its initial size, determined by the dichotomy criteria. Moreover, when spreading happens, we show that the expanding front of the epidemic has an asymptotic spreading speed, which is determined by an associated semi-wave problem.

2021, 26(4): 2239-2255 doi: 10.3934/dcdsb.2021007 +[Abstract](238) +[HTML](93) +[PDF](812.83KB)
Abstract:

In this paper, we first establish the existence of semi-traveling wave solutions to a diffusive generalized Holling-Tanner predator-prey model in which the functional response may depend on both the predator and prey populations. Then, by constructing the Lyapunov function, we apply the obtained result to show the existence of traveling wave solutions to the diffusive Holling-Tanner predator-prey models with various functional responses, including the Lotka-Volterra type functional response, the Holling type Ⅱ functional response and the Beddington-DeAngelis functional response.

2021, 26(4): 2257-2272 doi: 10.3934/dcdsb.2020212 +[Abstract](612) +[HTML](315) +[PDF](597.03KB)
Abstract:

In this paper, we investigate an in-host model for the viral dynamics of HIV-1 infection and its interaction with the CTL immune response. The model is sufficiently general to allow nonlinear forms for both viral infection and CTL response. Threshold parameters are identified that completely determine the global dynamics and outcomes of the virus-target cell-CTL interactions. Impacts of key parameter values for CTL functions and viral budding rate on the HIV-1 viral load and CD4 count are investigated using numerical simulations. Results support clinical evidence for important differences between HIV-1 nonprogressors and progressors.

2021, 26(4): 2273-2297 doi: 10.3934/dcdsb.2021006 +[Abstract](194) +[HTML](93) +[PDF](648.31KB)
Abstract:

The effect of competition is an important topic in spatial ecology. This paper deals with a general two-species competition system in open advective and inhomogeneous environments. At first, the critical values on the interspecific competition coefficients are established, which determine the stability of semi-trivial steady states. Secondly, by analyzing the nonexistence of coexistence steady states and using the theory of monotone dynamical system, we find that the competitive exclusion principle holds if one of the interspecific competition coefficients is large and the other is in a certain range. Thirdly, in terms of these critical values, the structure and direction of bifurcating branches of positive equilibria arising from two semi-trivial steady states are given by means of the bifurcation theory and stability analysis. Finally, we show that multiple coexistence occurs under certain regimes.

2021, 26(4): 2299-2322 doi: 10.3934/dcdsb.2020324 +[Abstract](282) +[HTML](152) +[PDF](627.17KB)
Abstract:

Tuberculosis infection is still a major threat to humans and it may progress slowly or rapidly to clearance, latent infection, or active disease. In this paper, considering T cells can perform acceleration effect on their own recruitment, an in-host model of Mycobacterium tuberculosis is studied. Focus type and elliptic type of nilpotent singularities of codimension 3 are analyzed in this four dimensional model. Complex dynamical behaviors such as homoclinic loop, saddle-node bifurcation of limit cycle and co-existence of two limit cycles are revealed by bifurcation analysis. Especially, the slow-fast periodic solution with large-amplitude or small-amplitude is observed in numerical simulations, which provides a perfect explanation for the reactivation of latent infection.

2021, 26(4): 2323-2342 doi: 10.3934/dcdsb.2021046 +[Abstract](120) +[HTML](54) +[PDF](1083.12KB)
Abstract:

This paper is concerned with the asymptotic stability of wave fronts and oscillatory waves for some predator-prey models. By spectral analysis and applying Evans function method with some numerical simulations, we show that the two types of waves with noncritical speeds are spectrally stable and nonlinearly exponentially stable in some exponentially weighted spaces.

2019  Impact Factor: 1.27