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Discrete & Continuous Dynamical Systems - B

January 2021 , Volume 26 , Issue 1

20 years of DCDS-B

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Preface for the special issue "20 years of DCDS-B"
Peter E. Kloeden and Yuan Lou
2021, 26(1): i-ii doi: 10.3934/dcdsb.2020372 +[Abstract](644) +[HTML](191) +[PDF](85.05KB)
Abstract:
Bifurcations in periodic integrodifference equations in $ C(\Omega) $ I: Analytical results and applications
Christian Aarset and Christian Pötzsche
2021, 26(1): 1-60 doi: 10.3934/dcdsb.2020231 +[Abstract](672) +[HTML](249) +[PDF](2674.93KB)
Abstract:

We study local bifurcations of periodic solutions to time-periodic (systems of) integrodifference equations over compact habitats. Such infinite-dimensional discrete dynamical systems arise in theoretical ecology as models to describe the spatial dispersal of species having nonoverlapping generations. Our explicit criteria allow us to identify branchings of fold- and crossing curve-type, which include the classical transcritical-, pitchfork- and flip-scenario as special cases. Indeed, not only tools to detect qualitative changes in models from e.g. spatial ecology and related simulations are provided, but these critical transitions are also classified. In addition, the bifurcation behavior of various time-periodic integrodifference equations is investigated and illustrated. This requires a combination of analytical methods and numerical tools based on Nyström discretization of the integral operators involved.

An adaptive finite element DtN method for the three-dimensional acoustic scattering problem
Gang Bao, Mingming Zhang, Bin Hu and Peijun Li
2021, 26(1): 61-79 doi: 10.3934/dcdsb.2020351 +[Abstract](399) +[HTML](84) +[PDF](484.38KB)
Abstract:

This paper is concerned with a numerical solution of the acoustic scattering by a bounded impenetrable obstacle in three dimensions. The obstacle scattering problem is formulated as a boundary value problem in a bounded domain by using a Dirichlet-to-Neumann (DtN) operator. An a posteriori error estimate is derived for the finite element method with the truncated DtN operator. The a posteriori error estimate consists of the finite element approximation error and the truncation error of the DtN operator, where the latter is shown to decay exponentially with respect to the truncation parameter. Based on the a posteriori error estimate, an adaptive finite element method is developed for the obstacle scattering problem. The truncation parameter is determined by the truncation error of the DtN operator and the mesh elements for local refinement are marked through the finite element approximation error. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

A $ C^1 $ Petrov-Galerkin method and Gauss collocation method for 1D general elliptic problems and superconvergence
Waixiang Cao, Lueling Jia and Zhimin Zhang
2021, 26(1): 81-105 doi: 10.3934/dcdsb.2020327 +[Abstract](332) +[HTML](92) +[PDF](426.5KB)
Abstract:

In this paper, we present and study \begin{document}$ C^1 $\end{document} Petrov-Galerkin and Gauss collocation methods with arbitrary polynomial degree \begin{document}$ k $\end{document} (\begin{document}$ \ge 3 $\end{document}) for one-dimen\-sional elliptic equations. We prove that, the solution and its derivative approximations converge with rate \begin{document}$ 2k-2 $\end{document} at all grid points; and the solution approximation is superconvergent at all interior roots of a special Jacobi polynomial of degree \begin{document}$ k+1 $\end{document} in each element, the first-order derivative approximation is superconvergent at all interior \begin{document}$ k-2 $\end{document} Lobatto points, and the second-order derivative approximation is superconvergent at \begin{document}$ k-1 $\end{document} Gauss points, with an order of \begin{document}$ k+2 $\end{document}, \begin{document}$ k+1 $\end{document}, and \begin{document}$ k $\end{document}, respectively. As a by-product, we prove that both the Petrov-Galerkin solution and the Gauss collocation solution are superconvergent towards a particular Jacobi projection of the exact solution in \begin{document}$ H^2 $\end{document}, \begin{document}$ H^1 $\end{document}, and \begin{document}$ L^2 $\end{document} norms. All theoretical findings are confirmed by numerical experiments.

Dynamics of a discrete-time stoichiometric optimal foraging model
Ming Chen and Hao Wang
2021, 26(1): 107-120 doi: 10.3934/dcdsb.2020264 +[Abstract](660) +[HTML](184) +[PDF](684.79KB)
Abstract:

In this paper, we discretize and analyze a stoichiometric optimal foraging model where the grazer's feeding effort depends on the producer's nutrient quality. We systematically make comparisons of the dynamical behaviors between the discrete-time model and the continuous-time model to study the robustness of model predictions to time discretization. When the maximum growth rate of producer is low, both model types admit similar dynamics including bistability and deterministic extinction of the grazer caused by low nutrient quality of the producer. Especially, the grazer is benefited from optimal foraging similarly in both discrete-time and continuous-time models. When the maximum growth rate of producer is high, dynamics of the discrete-time model are more complex including chaos. A phenomenal observation is that under extremely high light intensities, the grazer in the continuous-time model tends to perish due to poor food quality, however, the grazer in the discrete-time model persists in regular or irregular oscillatory ways. This significant difference indicates the necessity of studying discrete-time models which naturally include species' generations and are thus more popular in theoretical biology. Finally, we discuss how the shape of the quality-based feeding function regulates the beneficial or restraint effect of optimal foraging on the grazer population.

Bilinear equations in Hilbert space driven by paths of low regularity
Petr Čoupek and María J. Garrido-Atienza
2021, 26(1): 121-154 doi: 10.3934/dcdsb.2020230 +[Abstract](490) +[HTML](218) +[PDF](527.45KB)
Abstract:

In the article, some bilinear evolution equations in Hilbert space driven by paths of low regularity are considered and solved explicitly. The driving paths are scalar-valued and continuous, and they are assumed to have a finite \begin{document}$ p $\end{document}-th variation along a sequence of partitions in the sense given by Cont and Perkowski [Trans. Amer. Math. Soc. Ser. B, 6 (2019) 161–186] (\begin{document}$ p $\end{document} being an even positive integer). Typical functions that satisfy this condition are trajectories of the fractional Brownian motion with Hurst parameter \begin{document}$H=1 / p$\end{document}. A strong solution to the bilinear problem is shown to exist if there is a solution to a certain non–autonomous initial value problem. Subsequently, sufficient conditions for the existence of the solution to this initial value problem are given. The abstract results are applied to several stochastic partial differential equations with multiplicative fractional noise, both of the parabolic and hyperbolic type, that are solved explicitly in a pathwise sense.

A spatial food chain model for the Black Sea Anchovy, and its optimal fishery
Mahir Demir and Suzanne Lenhart
2021, 26(1): 155-171 doi: 10.3934/dcdsb.2020373 +[Abstract](284) +[HTML](78) +[PDF](2028.75KB)
Abstract:

We present a spatial food chain model on a bounded domain coupled with optimal control theory to examine harvesting strategies. Motivated by the fishery industry in the Black Sea, the anchovy stock and two more trophic levels are modeled using nonlinear parabolic partial differential equations with logistic growth, movement by diffusion and advection, and Neumann boundary conditions. Necessary conditions for the optimal harvesting control are established. The objective for the problem is to find the spatial optimal harvesting strategy that maximizes the discounted net value of the anchovy population. Numerical simulations using data from the Black Sea are presented. We discuss spatial features of harvesting and the effects of diffusion and advection (migration speed) on the anchovy population. We also present the landing of anchovy and its net profit by applying two different harvesting strategies.

The motion of weakly interacting localized patterns for reaction-diffusion systems with nonlocal effect
Shin-Ichiro Ei and Hiroshi Ishii
2021, 26(1): 173-190 doi: 10.3934/dcdsb.2020329 +[Abstract](462) +[HTML](115) +[PDF](747.17KB)
Abstract:

In this paper, we analyze the interaction of localized patterns such as traveling wave solutions for reaction-diffusion systems with nonlocal effect in one space dimension. We consider the case that a nonlocal effect is given by the convolution with a suitable integral kernel. At first, we deduce the equation describing the movement of interacting localized patterns in a mathematically rigorous way, assuming that there exists a linearly stable localized solution for general reaction-diffusion systems with nonlocal effect. When the distances between localized patterns are sufficiently large, the motion of localized patterns can be reduced to the equation for the distances between them. Finally, using this equation, we analyze the interaction of front solutions to some nonlocal scalar equation. Under some assumptions, we can show that the front solutions are interacting attractively for a large class of integral kernels.

Chaotic dynamics in a simple predator-prey model with discrete delay
Guihong Fan and Gail S. K. Wolkowicz
2021, 26(1): 191-216 doi: 10.3934/dcdsb.2020263 +[Abstract](546) +[HTML](219) +[PDF](5405.39KB)
Abstract:

A discrete delay is included to model the time between the capture of the prey and its conversion to viable biomass in the simplest classical Gause type predator-prey model that has equilibrium dynamics without delay. As the delay increases from zero, the coexistence equilibrium undergoes a supercritical Hopf bifurcation, two saddle-node bifurcations of limit cycles, and a cascade of period doublings, eventually leading to chaos. The resulting periodic orbits and the strange attractor resemble their counterparts for the Mackey-Glass equation. Due to the global stability of the system without delay, this complicated dynamics can be solely attributed to the introduction of the delay. Since many models include predator-prey like interactions as submodels, this study emphasizes the importance of understanding the implications of overlooking delay in such models on the reliability of the model-based predictions, especially since temperature is known to have an effect on the length of certain delays.

Predator – Prey/Host – Parasite: A fragile ecoepidemic system under homogeneous infection incidence
Alex P. Farrell and Horst R. Thieme
2021, 26(1): 217-267 doi: 10.3934/dcdsb.2020328 +[Abstract](315) +[HTML](155) +[PDF](541.79KB)
Abstract:

To underpin the concern that environmental change can flip an ecosystem from stable persistence to sudden total collapse, we consider a class of so-called ecoepidemic models, predator – prey/host – parasite systems, in which a base species is prey to a predator species and host to a micro-parasite species. Our model uses generalized frequency-dependent incidence for the disease transmission and mass action kinetics for predation.

We show that a large variety of dynamics can arise, ranging from dynamic persistence of all three species to either total ecosystem collapse caused by high transmissibility of the parasite on the one hand or to parasite extinction and prey-predator survival due to low parasite transmissibility on the other hand. We identify a threshold parameter (tipping number) for the transition of the ecosystem from uniform prey/host persistence to total extinction under suitable initial conditions.

Coexistence of competing consumers on a single resource in a hybrid model
Yunfeng Geng, Xiaoying Wang and Frithjof Lutscher
2021, 26(1): 269-297 doi: 10.3934/dcdsb.2020140 +[Abstract](911) +[HTML](327) +[PDF](1587.09KB)
Abstract:

The question of whether and how two competing consumers can coexist on a single limiting resource has a long tradition in ecological theory. We build on a recent seasonal (hybrid) model for one consumer and one resource, and we extend it by introducing a second consumer. Consumers reproduce only once per year, the resource reproduces throughout the"summer" season. When we use linear consumer reproduction between years, we find explicit expressions for the trivial and semi-trivial equilibria, and we prove that there is no positive equilibrium generically. When we use non-linear consumer reproduction, we determine conditions for which both semi-trivial equilibria are unstable. We prove that a unique positive equilibrium exists in this case, and we find an explicit analytical expression for it. By linear analysis and numerical simulation, we find bifurcations from the stable equilibrium to population cycles that may appear through period-doubling or Hopf bifurcations. We interpret our results in terms of climate change that changes the length of the"summer" season.

Computing complete Lyapunov functions for discrete-time dynamical systems
Peter Giesl, Zachary Langhorne, Carlos Argáez and Sigurdur Hafstein
2021, 26(1): 299-336 doi: 10.3934/dcdsb.2020331 +[Abstract](362) +[HTML](114) +[PDF](28240.81KB)
Abstract:

A complete Lyapunov function characterizes the behaviour of a general discrete-time dynamical system. In particular, it divides the state space into the chain-recurrent set where the complete Lyapunov function is constant along trajectories and the part where the flow is gradient-like and the complete Lyapunov function is strictly decreasing along solutions. Moreover, the level sets of a complete Lyapunov function provide information about attractors, repellers, and basins of attraction.

We propose two novel classes of methods to compute complete Lyapunov functions for a general discrete-time dynamical system given by an iteration. The first class of methods computes a complete Lyapunov function by approximating the solution of an ill-posed equation for its discrete orbital derivative using meshfree collocation. The second class of methods computes a complete Lyapunov function as solution of a minimization problem in a reproducing kernel Hilbert space. We apply both classes of methods to several examples.

The Navier-Stokes-Cahn-Hilliard equations for mildly compressible binary fluid mixtures
Andrea Giorgini, Roger Temam and Xuan-Truong Vu
2021, 26(1): 337-366 doi: 10.3934/dcdsb.2020141 +[Abstract](886) +[HTML](381) +[PDF](457.03KB)
Abstract:

We study the well-posedness for the mildly compressible Navier-Stokes-Cahn-Hilliard system with non-constant viscosity and Landau potential in two and three dimensional domains.

Ecological and evolutionary dynamics in advective environments: Critical domain size and boundary conditions
Wenrui Hao, King-Yeung Lam and Yuan Lou
2021, 26(1): 367-400 doi: 10.3934/dcdsb.2020283 +[Abstract](502) +[HTML](188) +[PDF](1306.88KB)
Abstract:

We consider the ecological and evolutionary dynamics of a reaction-diffusion-advection model for populations residing in a one-dimensional advective homogeneous environment, with emphasis on the effects of boundary conditions and domain size. We assume that there is a net loss of individuals at the downstream end with rate \begin{document}$ b \geq 0 $\end{document}, while the no-flux condition is imposed on the upstream end. For the single species model, it is shown that the critical patch size is a decreasing function of the dispersal rate when \begin{document}$ b \leq 3/2 $\end{document}; whereas it first decreases and then increases when \begin{document}$ b >3/2 $\end{document}.

For the two-species competition model, we show that the infinite dispersal rate is evolutionarily stable for \begin{document}$ b < 3/2 $\end{document} and, when dispersal rates of both species are large, the population with larger dispersal rate always displaces the population with the smaller rate. For certain specific population loss rate \begin{document}$ b<3/2 $\end{document}, it is also shown that there can be up to three evolutionarily stable strategies. For \begin{document}$ b>3/2 $\end{document}, it is proved that the infinite random dispersal rate is not evolutionarily stable, and that, for some specific \begin{document}$ b>3/2 $\end{document}, a finite dispersal rate is evolutionarily stable. Furthermore, for the intermediate domain size, this dispersal rate is optimal in the sense that the species adopting this rate is able to displace its competitor with a similar but different rate. Finally, nine qualitatively different pairwise invasibility plots are obtained by varying the parameter \begin{document}$ b $\end{document} and the domain size.

On a singularly perturbed semi-linear problem with Robin boundary conditions
Qianqian Hou, Tai-Chia Lin and Zhi-An Wang
2021, 26(1): 401-414 doi: 10.3934/dcdsb.2020083 +[Abstract](1148) +[HTML](439) +[PDF](361.26KB)
Abstract:

This paper is concerned with a semi-linear elliptic problem with Robin boundary condition:

where \begin{document}$ \Omega \subset {\mathbb R}^N (N\geq 1) $\end{document} is a bounded domain with smooth boundary, \begin{document}$ \vec{n} $\end{document} denotes the unit outward normal vector of \begin{document}$ \partial \Omega $\end{document} and \begin{document}$ \gamma \in {\mathbb R}/\{0\} $\end{document}. \begin{document}$ \varepsilon $\end{document} and \begin{document}$ \lambda $\end{document} are positive constants. The problem (*) is derived from the well-known singular Keller-Segel system. When \begin{document}$ \gamma>0 $\end{document}, we show there is only trivial solution \begin{document}$ w = 0 $\end{document}. When \begin{document}$ \gamma<0 $\end{document} and \begin{document}$ \Omega = B_R(0) $\end{document} is a ball, we show that problem (*) has a non-constant solution which converges to zero uniformly as \begin{document}$ \varepsilon $\end{document} tends to zero. The main idea of this paper is to transform the Robin problem (*) to a nonlocal Dirichelt problem by a Cole-Hopf type transformation and then use the shooting method to obtain the existence of the transformed nonlocal Dirichlet problem. With the results for (*), we get the existence of non-constant stationary solutions to the original singular Keller-Segel system.

The dynamics of a two host-two virus system in a chemostat environment
Sze-Bi Hsu and Yu Jin
2021, 26(1): 415-441 doi: 10.3934/dcdsb.2020298 +[Abstract](382) +[HTML](138) +[PDF](865.68KB)
Abstract:

The coevolution or coexistence of multiple viruses with multiple hosts has been an important issue in viral ecology. This paper is to study the mathematical properties of the solutions of a chemostat model for two host species and two virus species. By virtue of the global dynamics of its submodels and the theories of uniform persistence and Hopf bifurcation, we derive sufficient conditions for the coexistence of two hosts with two viruses and coexistence of two hosts with one virus, as well as occurrence of Hopf bifurcation.

Modeling multiple taxis: Tumor invasion with phenotypic heterogeneity, haptotaxis, and unilateral interspecies repellence
Niklas Kolbe, Nikolaos Sfakianakis, Christian Stinner, Christina Surulescu and Jonas Lenz
2021, 26(1): 443-481 doi: 10.3934/dcdsb.2020284 +[Abstract](518) +[HTML](163) +[PDF](9158.09KB)
Abstract:

We provide a short review of existing models with multiple taxis performed by (at least) one species and consider a new mathematical model for tumor invasion featuring two mutually exclusive cell phenotypes (migrating and proliferating). The migrating cells perform nonlinear diffusion and two types of taxis in response to non-diffusing cues: away from proliferating cells and up the gradient of surrounding tissue. Transitions between the two cell subpopulations are influenced by subcellular (receptor binding) dynamics, thus conferring the setting a multiscale character.

We prove global existence of weak solutions to a simplified model version and perform numerical simulations for the full setting under several phenotype switching and motility scenarios. We also compare (via simulations) this model with the corresponding haptotaxis-chemotaxis one featuring indirect chemorepellent production and provide a discussion about possible model extensions and mathematical challenges.

On the role of pharmacometrics in mathematical models for cancer treatments
Urszula Ledzewicz and Heinz Schättler
2021, 26(1): 483-499 doi: 10.3934/dcdsb.2020213 +[Abstract](593) +[HTML](237) +[PDF](464.36KB)
Abstract:

We review and discuss various aspects that the modeling of pharmacometric properties has on the structure of optimal solutions in mathematical models for cancer treatment. These include (i) the changes in the interpretation of the solutions as pharmacokinetic (PK) models are added, respectively deleted from the modeling and (ii) qualitative changes in the structures of optimal controls that occur as pharmacodynamic (PD) models are varied. The results will be illustrated with a sample of models for cancer treatment.

Topological phase transition III: Solar surface eruptions and sunspots
Tian Ma and Shouhong Wang
2021, 26(1): 501-514 doi: 10.3934/dcdsb.2020350 +[Abstract](285) +[HTML](86) +[PDF](490.01KB)
Abstract:

This paper is aimed to provide a new theory for the formation of the solar surface eruptions and sunspots. The key ingredient of the study is the new anti-diffusive effect of heat, based on the recently developed statistical theory of heat by the authors [4]. The anti-diffusive effect of heat states that due to the higher rate of photon absorption and emission of the particles with higher energy levels, the photon flux will move toward to the higher temperature regions from the lower temperature regions. This anti-diffusive effect of heat leads to a modified law of heat transfer, which includes a reversed heat flux counteracting the heat diffusion. It is this anti-diffusive effect of heat and thereby the modified law of heat transfer that lead to the temperature blow-up and consequently the formation of sunspots, solar eruptions, and solar prominences. This anti-diffusive effect of heat may be utilized to design a plasma instrument, directly converting solar energy into thermal energy. This may likely offer a new form of fuel much more efficient than the photovoltaic devices.

Rich dynamics of a simple delay host-pathogen model of cell-to-cell infection for plant virus
Tin Phan, Bruce Pell, Amy E. Kendig, Elizabeth T. Borer and Yang Kuang
2021, 26(1): 515-539 doi: 10.3934/dcdsb.2020261 +[Abstract](570) +[HTML](203) +[PDF](601.38KB)
Abstract:

Viral dynamics within plant hosts can be important for understanding plant disease prevalence and impacts. However, few mathematical modeling efforts aim to characterize within-plant viral dynamics. In this paper, we derive a simple system of delay differential equations that describes the spread of infection throughout the plant by barley and cereal yellow dwarf viruses via the cell-to-cell mechanism. By incorporating ratio-dependent incidence function and logistic growth of the healthy cells, the model can capture a wide range of biologically relevant phenomena via the disease-free, endemic, mutual extinction steady states, and a stable periodic orbit. We show that when the basic reproduction number is less than \begin{document}$ 1 $\end{document} (\begin{document}$ R_0 < 1 $\end{document}), the disease-free steady state is asymptotically stable. When \begin{document}$ R_0>1 $\end{document}, the dynamics either converge to the endemic equilibrium or enter a periodic orbit. Using a ratio-dependent transformation, we show that if the infection rate is very high relative to the growth rate of healthy cells, then the system collapses to the mutual extinction steady state. Numerical and bifurcation simulations are provided to demonstrate our theoretical results. Finally, we carry out parameter estimation using experimental data to characterize the effects of varying nutrients on the dynamics of the system. Our parameter estimates suggest that varying the nutrient supply of nitrogen and phosphorous can alter the dynamics of the infection in plants, specifically reducing the rate of viral production and the rate of infection in certain cases.

Nonlinear dynamics in tumor-immune system interaction models with delays
Shigui Ruan
2021, 26(1): 541-602 doi: 10.3934/dcdsb.2020282 +[Abstract](540) +[HTML](167) +[PDF](1287.75KB)
Abstract:

In this paper, we review some recent results on the nonlinear dynamics of delayed differential equation models describing the interaction between tumor cells and effector cells of the immune system, in which the delays represent times necessary for molecule production, proliferation, differentiation of cells, transport, etc. First we consider a tumor-immune system interaction model with a single delay and present results on the existence and local stability of equilibria as well as the existence of Hopf bifurcation in the model when the delay varies. Second we investigate a tumor-immune system interaction model with two delays and show that the model undergoes various possible bifurcations including Hopf, Bautin, Fold-Hopf (zero-Hopf), and Hopf-Hopf bifurcations. Finally we discuss a tumor-immune system interaction model with three delays and demonstrate that the model exhibits more complex behaviors including chaos. Numerical simulations are provided to illustrate the nonlinear dynamics of the delayed tumor-immune system interaction models. More interesting issues and questions on modeling and analyzing tumor-immune dynamics are given in the discussion section.

Equilibrium validation in models for pattern formation based on Sobolev embeddings
Evelyn Sander and Thomas Wanner
2021, 26(1): 603-632 doi: 10.3934/dcdsb.2020260 +[Abstract](481) +[HTML](166) +[PDF](1467.47KB)
Abstract:

In the study of equilibrium solutions for partial differential equations there are so many equilibria that one cannot hope to find them all. Therefore one usually concentrates on finding individual branches of equilibrium solutions. On the one hand, a rigorous theoretical understanding of these branches is ideal but not generally tractable. On the other hand, numerical bifurcation searches are useful but not guaranteed to give an accurate structure, in that they could miss a portion of a branch or find a spurious branch where none exists. In a series of recent papers, we have aimed for a third option. Namely, we have developed a method of computer-assisted proofs to prove both existence and isolation of branches of equilibrium solutions. In the current paper, we extend these techniques to the Ohta-Kawasaki model for the dynamics of diblock copolymers in dimensions one, two, and three, by giving a detailed description of the analytical underpinnings of the method. Although the paper concentrates on applying the method to the Ohta-Kawasaki model, the functional analytic approach and techniques can be generalized to other parabolic partial differential equations.

The $ P^* $ rule in the stochastic Holt-Lawton model of apparent competition
Sebastian J. Schreiber
2021, 26(1): 633-644 doi: 10.3934/dcdsb.2020374 +[Abstract](311) +[HTML](110) +[PDF](358.96KB)
Abstract:

In 1993, Holt and Lawton introduced a stochastic model of two host species parasitized by a common parasitoid species. We introduce and analyze a generalization of these stochastic difference equations with any number of host species, stochastically varying parasitism rates, stochastically varying host intrinsic fitnesses, and stochastic immigration of parasitoids. Despite the lack of direct, host density-dependence, we show that this system is dissipative i.e. enters a compact set in finite time for all initial conditions. When there is a single host species, stochastic persistence and extinction of the host is characterized using external Lyapunov exponents corresponding to the average per-capita growth rates of the host when rare. When a single host persists, say species \begin{document}$ i $\end{document}, a explicit expression is derived for the average density, \begin{document}$ P_i^* $\end{document}, of the parasitoid at the stationary distributions supporting both species. When there are multiple host species, we prove that the host species with the largest \begin{document}$ P_i^* $\end{document} value stochastically persists, while the other host species are asymptotically driven to extinction. A review of the main mathematical methods used to prove the results and future challenges are given.

Efficient and accurate sav schemes for the generalized Zakharov systems
Jie Shen and Nan Zheng
2021, 26(1): 645-666 doi: 10.3934/dcdsb.2020262 +[Abstract](393) +[HTML](206) +[PDF](1743.07KB)
Abstract:

We develop in this paper efficient and accurate numerical schemes based on the scalar auxiliary variable (SAV) approach for the generalized Zakharov system and generalized vector Zakharov system. These schemes are second-order in time, linear, unconditionally stable, only require solving linear systems with constant coefficients at each time step, and preserve exactly a modified Hamiltonian. Ample numerical results are presented to demonstrate the accuracy and robustness of the schemes.

Stationary solutions of a free boundary problem modeling the growth of vascular tumors with a necrotic core
Huijuan Song, Bei Hu and Zejia Wang
2021, 26(1): 667-691 doi: 10.3934/dcdsb.2020084 +[Abstract](1099) +[HTML](450) +[PDF](447.27KB)
Abstract:

In this paper, we present a rigorous mathematical analysis of a free boundary problem modeling the growth of a vascular solid tumor with a necrotic core. If the vascular system supplies the nutrient concentration \begin{document}$ \sigma $\end{document} to the tumor at a rate \begin{document}$ \beta $\end{document}, then \begin{document}$ \frac{\partial\sigma}{\partial\bf n}+\beta(\sigma-\bar\sigma) = 0 $\end{document} holds on the tumor boundary, where \begin{document}$ \bf n $\end{document} is the unit outward normal to the boundary and \begin{document}$ \bar\sigma $\end{document} is the nutrient concentration outside the tumor. The living cells in the nonnecrotic region proliferate at a rate \begin{document}$ \mu $\end{document}. We show that for any given \begin{document}$ \rho>0 $\end{document}, there exists a unique \begin{document}$ R\in(\rho, \infty) $\end{document} such that the corresponding radially symmetric solution solves the steady-state necrotic tumor system with necrotic core boundary \begin{document}$ r = \rho $\end{document} and outer boundary \begin{document}$ r = R $\end{document}; moreover, there exist a positive integer \begin{document}$ n^{**} $\end{document} and a sequence of \begin{document}$ \mu_n $\end{document}, symmetry-breaking stationary solutions bifurcate from the radially symmetric stationary solution for each \begin{document}$ \mu_n $\end{document} (even \begin{document}$ n\ge n^{**}) $\end{document}.

Random attractors for 2D stochastic micropolar fluid flows on unbounded domains
Wenlong Sun, Jiaqi Cheng and Xiaoying Han
2021, 26(1): 693-716 doi: 10.3934/dcdsb.2020189 +[Abstract](565) +[HTML](252) +[PDF](523.81KB)
Abstract:

The asymptotic behavior of a model for 2D incompressible stochastic micropolar fluid flows with rough noise on a Poincaré domain is investigated. First, the existence and uniqueness of solutions to an evolution equation arising from the underlying stochastic micropolar fluid model is established via the Galerkin method and energy method. Then the existence of a random attractor is studied by using the theory of random dynamical systems for which the noise is dealt with by appropriate reproducing kernel Hilbert space.

How to detect Wada basins
Alexandre Wagemakers, Alvar Daza and Miguel A. F. Sanjuán
2021, 26(1): 717-739 doi: 10.3934/dcdsb.2020330 +[Abstract](374) +[HTML](99) +[PDF](6379.1KB)
Abstract:

We present a review of the different techniques available to study a special kind of fractal basins of attraction known as Wada basins, which have the intriguing property of having a single boundary separating three or more basins. We expose several approaches to identify this topological property that rely on different, but not exclusive, definitions of the Wada property.

Global optimization-based dimer method for finding saddle points
Bing Yu and Lei Zhang
2021, 26(1): 741-753 doi: 10.3934/dcdsb.2020139 +[Abstract](857) +[HTML](386) +[PDF](1596.31KB)
Abstract:

Searching saddle points on the potential energy surface is a challenging problem in the rare event. When there exist multiple saddle points, sampling different initial guesses are needed in local search methods in order to find distinct saddle points. In this paper, we present a novel global optimization-based dimer method (GOD) to efficiently search saddle points by coupling ant colony optimization (ACO) algorithm with optimization-based shrinking dimer (OSD) method. In particular, we apply OSD method as a local search algorithm for saddle points and construct a pheromone function in ACO to update the global population. By applying a two-dimensional example and a benchmark problem of seven-atom island on the (111) surface of an FCC crystal, we demonstrate that GOD shows a significant improvement in computational efficiency compared with OSD method. Our algorithm is the first try to apply the global optimization technique in searching saddle points and it offers a new framework to open up possibilities of adopting other global optimization methods.

2019  Impact Factor: 1.27

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