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Discrete & Continuous Dynamical Systems - B
January 2016 , Volume 21 , Issue 1
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2016, 21(1): 1-11
doi: 10.3934/dcdsb.2016.21.1
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Abstract:
Using the averaging theory we study the periodic solutions and their linear stability of the $3$--dimensional chaotic quadratic polynomial differential systems without equilibria studied in [3]. All these differential systems depend only on one--parameter.
Using the averaging theory we study the periodic solutions and their linear stability of the $3$--dimensional chaotic quadratic polynomial differential systems without equilibria studied in [3]. All these differential systems depend only on one--parameter.
2016, 21(1): 13-35
doi: 10.3934/dcdsb.2016.21.13
+[Abstract](1355)
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Abstract:
We study a diffusive logistic equation with a free boundary in time-periodic environment. To understand the effect of the diffusion rate $d$, the original habitat radius $h_0$, the spreading capability $\mu$ , and the initial density $u_0$ on the dynamics of the problem, we divide the time-periodic habitat into two cases: favorable habitat and unfavorable habitat. By choosing $d, h_0, \mu$ and $u_0$ as variable parameters, we obtain a spreading-vanishing dichotomy and sharp criteria for the spreading and vanishing in time-periodic environment. We introduce the principal eigenvalue $\lambda_1(d, \alpha-\gamma, h_{0}, T)$ to determine the spreading and vanishing of the invasive species. We prove that if $\lambda_1(d, \alpha-\gamma, h_0, T)\leq 0$, the spreading must happen; while if $\lambda_1(d, \alpha-\gamma, h_0, T)>0$, the spreading is also possible. Our results show that the species in the favorable habitat can establish itself if the diffusion rate is small or the occupying habitat is large. In an unfavorable habitat, the species vanishes if the initial density of the species is small. Moreover, when spreading occurs, the asymptotic spreading speed of the free boundary is determined.
We study a diffusive logistic equation with a free boundary in time-periodic environment. To understand the effect of the diffusion rate $d$, the original habitat radius $h_0$, the spreading capability $\mu$ , and the initial density $u_0$ on the dynamics of the problem, we divide the time-periodic habitat into two cases: favorable habitat and unfavorable habitat. By choosing $d, h_0, \mu$ and $u_0$ as variable parameters, we obtain a spreading-vanishing dichotomy and sharp criteria for the spreading and vanishing in time-periodic environment. We introduce the principal eigenvalue $\lambda_1(d, \alpha-\gamma, h_{0}, T)$ to determine the spreading and vanishing of the invasive species. We prove that if $\lambda_1(d, \alpha-\gamma, h_0, T)\leq 0$, the spreading must happen; while if $\lambda_1(d, \alpha-\gamma, h_0, T)>0$, the spreading is also possible. Our results show that the species in the favorable habitat can establish itself if the diffusion rate is small or the occupying habitat is large. In an unfavorable habitat, the species vanishes if the initial density of the species is small. Moreover, when spreading occurs, the asymptotic spreading speed of the free boundary is determined.
2016, 21(1): 37-53
doi: 10.3934/dcdsb.2016.21.37
+[Abstract](1601)
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Abstract:
In this study, we investigate a Hindmarsh-Rose-type model with the structure of recurrent neural feedback. The number of equilibria and their stability for the model with zero delay are reviewed first. We derive conditions for the existence of a Hopf bifurcation in the model and derive equations for the direction and stability of the bifurcation with delay as the bifurcation parameter. The ranges of parameter values for the existence of a Hopf bifurcation and the system responses with various levels of delay are obtained. When a Hopf bifurcation due to delay occurs, canard-like mixed-mode oscillations (MMOs) are produced at the parameter value for which either the fold bifurcation of cycles or homoclinic bifurcation occurs in the system without delay. This behavior can be found in a planar system with delay but not in a planar system without delay. Therefore, the results of this study will be helpful for determining suitable parameters to represent MMOs with a simple system with delay.
In this study, we investigate a Hindmarsh-Rose-type model with the structure of recurrent neural feedback. The number of equilibria and their stability for the model with zero delay are reviewed first. We derive conditions for the existence of a Hopf bifurcation in the model and derive equations for the direction and stability of the bifurcation with delay as the bifurcation parameter. The ranges of parameter values for the existence of a Hopf bifurcation and the system responses with various levels of delay are obtained. When a Hopf bifurcation due to delay occurs, canard-like mixed-mode oscillations (MMOs) are produced at the parameter value for which either the fold bifurcation of cycles or homoclinic bifurcation occurs in the system without delay. This behavior can be found in a planar system with delay but not in a planar system without delay. Therefore, the results of this study will be helpful for determining suitable parameters to represent MMOs with a simple system with delay.
2016, 21(1): 55-65
doi: 10.3934/dcdsb.2016.21.55
+[Abstract](1357)
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Abstract:
The paper is concerned with a class of stochastic reaction-diffusion equations arising from a spatial population growth model in random environments. Under some sufficient conditions, Theorem 3.1 shows that the equation has a unique positive global solution in space $H^1(D)$. Then it is proven in Theorem 4.1 that the solution, as the population size, is ultimately bounded in the mean $L^2-$norm as the time tends to infinity. An almost-sure upper bound is also obtained for the long run time-average of the exponential rate of the population growth in $L^2-$norm together with the $L^p-$moment of the population size with $p \geq 2.$ It is also shown in Theorem 4.3 that there is a unique invariant measure that leads to a stationary population distribution. For illustration, an example is given.
The paper is concerned with a class of stochastic reaction-diffusion equations arising from a spatial population growth model in random environments. Under some sufficient conditions, Theorem 3.1 shows that the equation has a unique positive global solution in space $H^1(D)$. Then it is proven in Theorem 4.1 that the solution, as the population size, is ultimately bounded in the mean $L^2-$norm as the time tends to infinity. An almost-sure upper bound is also obtained for the long run time-average of the exponential rate of the population growth in $L^2-$norm together with the $L^p-$moment of the population size with $p \geq 2.$ It is also shown in Theorem 4.3 that there is a unique invariant measure that leads to a stationary population distribution. For illustration, an example is given.
2016, 21(1): 67-80
doi: 10.3934/dcdsb.2016.21.67
+[Abstract](1527)
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This paper concerns with the stability of the orbits for nearly integrable Hamiltonian systems. Based on Nekehoroshev's original works in [14], we present the definition of quasi-effective stability and prove a theorem on quasi-effective stability under the Rüssmann's non-degeneracy. Our result gives a relation between KAM theorem and effective stability. A rapidly converging iteration procedure with two parameters is designed.
This paper concerns with the stability of the orbits for nearly integrable Hamiltonian systems. Based on Nekehoroshev's original works in [14], we present the definition of quasi-effective stability and prove a theorem on quasi-effective stability under the Rüssmann's non-degeneracy. Our result gives a relation between KAM theorem and effective stability. A rapidly converging iteration procedure with two parameters is designed.
2016, 21(1): 81-102
doi: 10.3934/dcdsb.2016.21.81
+[Abstract](1926)
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Abstract:
This paper is concerned with the parabolic-elliptic Keller-Segel system with signal-dependent sensitivity $\chi(v)$, \begin{align*} \begin{cases} u_t=\Delta u - \nabla \cdot ( u \nabla \chi(v)) &\mathrm{in}\ \Omega\times(0,\infty), \\ 0=\Delta v -v+u &\mathrm{in}\ \Omega\times(0,\infty), \end{cases} \end{align*} under homogeneous Neumann boundary condition in a smoothly bounded domain $\Omega \subset \mathbb{R}^2$ with nonnegative initial data $u_0 \in C^{0}(\overline{\Omega})$, $\not\equiv 0$.
In the special case $\chi(v)=\chi_0 \log v\, (\chi_0>0)$, global existence and boundedness of the solution to the system were proved under some smallness condition on $\chi_0$ by Biler (1999) and Fujie, Winkler and Yokota (2015). In the present work, global existence and boundedness in the system will be established for general sensitivity $\chi$ satisfying $\chi'>0$ and $\chi'(s) \to 0 $ as $s\to \infty$. In particular, this establishes global existence and boundedness in the case $\chi(v)=\chi_0\log v$ with large $\chi_0>0$. Moreover, although the methods in the previous results are effective for only few specific cases, the present method can be applied to more general cases requiring only the essential conditions. Actually, our condition is necessary, since there are many radial blow-up solutions in the case $\inf_{s>0} \chi^\prime (s) >0$.
This paper is concerned with the parabolic-elliptic Keller-Segel system with signal-dependent sensitivity $\chi(v)$, \begin{align*} \begin{cases} u_t=\Delta u - \nabla \cdot ( u \nabla \chi(v)) &\mathrm{in}\ \Omega\times(0,\infty), \\ 0=\Delta v -v+u &\mathrm{in}\ \Omega\times(0,\infty), \end{cases} \end{align*} under homogeneous Neumann boundary condition in a smoothly bounded domain $\Omega \subset \mathbb{R}^2$ with nonnegative initial data $u_0 \in C^{0}(\overline{\Omega})$, $\not\equiv 0$.
In the special case $\chi(v)=\chi_0 \log v\, (\chi_0>0)$, global existence and boundedness of the solution to the system were proved under some smallness condition on $\chi_0$ by Biler (1999) and Fujie, Winkler and Yokota (2015). In the present work, global existence and boundedness in the system will be established for general sensitivity $\chi$ satisfying $\chi'>0$ and $\chi'(s) \to 0 $ as $s\to \infty$. In particular, this establishes global existence and boundedness in the case $\chi(v)=\chi_0\log v$ with large $\chi_0>0$. Moreover, although the methods in the previous results are effective for only few specific cases, the present method can be applied to more general cases requiring only the essential conditions. Actually, our condition is necessary, since there are many radial blow-up solutions in the case $\inf_{s>0} \chi^\prime (s) >0$.
2016, 21(1): 103-119
doi: 10.3934/dcdsb.2016.21.103
+[Abstract](2127)
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In this paper, a class of delay differential equations model of HIV infection dynamics with nonlinear transmissions and apoptosis induced by infected cells is proposed, and then the global properties of the model are considered. It shows that the infection-free equilibrium of the model is globally asymptotically stable if the basic reproduction number $R_{0}<1$, and globally attractive if $R_{0}=1$. The positive equilibrium of the model is locally asymptotically stable if $R_{0}>1$. Furthermore, it also shows that the model is permanent, and some explicit expressions for the eventual lower bounds of positive solutions of the model are given.
In this paper, a class of delay differential equations model of HIV infection dynamics with nonlinear transmissions and apoptosis induced by infected cells is proposed, and then the global properties of the model are considered. It shows that the infection-free equilibrium of the model is globally asymptotically stable if the basic reproduction number $R_{0}<1$, and globally attractive if $R_{0}=1$. The positive equilibrium of the model is locally asymptotically stable if $R_{0}>1$. Furthermore, it also shows that the model is permanent, and some explicit expressions for the eventual lower bounds of positive solutions of the model are given.
2016, 21(1): 121-131
doi: 10.3934/dcdsb.2016.21.121
+[Abstract](1541)
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Abstract:
We classify the global phase portraits in the Poincaré disc of the differential systems $\dot{x}=-y+xf(x,y),$ $\dot{y}=x+yf(x,y)$, where $f(x,y)$ is a homogeneous polynomial of degree 3. These systems have a uniform isochronous center at the origin. This paper together with the results presented in [9] completes the classification of the global phase portraits in the Poincaré disc of all quartic polynomial differential systems with a uniform isochronous center at the origin.
We classify the global phase portraits in the Poincaré disc of the differential systems $\dot{x}=-y+xf(x,y),$ $\dot{y}=x+yf(x,y)$, where $f(x,y)$ is a homogeneous polynomial of degree 3. These systems have a uniform isochronous center at the origin. This paper together with the results presented in [9] completes the classification of the global phase portraits in the Poincaré disc of all quartic polynomial differential systems with a uniform isochronous center at the origin.
2016, 21(1): 133-149
doi: 10.3934/dcdsb.2016.21.133
+[Abstract](1718)
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In this paper, a delayed viral infection model with nonlinear immune response and general incidence rate is discussed. We prove the existence and uniqueness of the equilibria. We study the effect of three kinds of time delays on the dynamics of the model. By using the Lyapunov functional and LaSalle invariance principle, we obtain the conditions of global stabilities of the infection-free equilibrium, the immune-exhausted equilibrium and the endemic equilibrium. It is shown that an increase of the viral-infection delay and the virus-production delay may stabilize the infection-free equilibrium, but the immune response delay can destabilize the equilibrium, leading to Hopf bifurcations. Numerical simulations are given to verify the analytical results. This can provide a possible interpretation for the viral oscillation observed in chronic hepatitis B virus (HBV) and human immunodeficiency virus (HIV) infected patients.
In this paper, a delayed viral infection model with nonlinear immune response and general incidence rate is discussed. We prove the existence and uniqueness of the equilibria. We study the effect of three kinds of time delays on the dynamics of the model. By using the Lyapunov functional and LaSalle invariance principle, we obtain the conditions of global stabilities of the infection-free equilibrium, the immune-exhausted equilibrium and the endemic equilibrium. It is shown that an increase of the viral-infection delay and the virus-production delay may stabilize the infection-free equilibrium, but the immune response delay can destabilize the equilibrium, leading to Hopf bifurcations. Numerical simulations are given to verify the analytical results. This can provide a possible interpretation for the viral oscillation observed in chronic hepatitis B virus (HBV) and human immunodeficiency virus (HIV) infected patients.
2016, 21(1): 151-172
doi: 10.3934/dcdsb.2016.21.151
+[Abstract](1500)
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We consider Cauchy problem for the semilinear plate equation with nonlocal nonlinearity. Under mild conditions on the damping coefficient, we prove that the semigroup generated by this problem possesses a global attractor.
We consider Cauchy problem for the semilinear plate equation with nonlocal nonlinearity. Under mild conditions on the damping coefficient, we prove that the semigroup generated by this problem possesses a global attractor.
2016, 21(1): 173-184
doi: 10.3934/dcdsb.2016.21.173
+[Abstract](1441)
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The purpose of this paper is to address synchronous chaos on the Julia set of complex-valued coupled map lattices (CCMLs). Our main results contain the following. First, we solve an inf min max problem for which its solution gives the fastest synchronized rate in a certain class of coupling matrices. Specifically, we show that for the class of real circulant matrices of size $4$, the coupling weights, possible complex numbers, yielding the fastest synchronized rate can be exactly obtained. Second, for individual map of the form $f_c(z)= z^2+ c$ with $|c|< \frac{1}{4}$, we show that the corresponding CCMLs acquires global synchrony on its Julia set with the number of the oscillators being $3$ or $4$ for the diffusive coupling. For $c=0$ and $-2$, the corresponding CCMLs obtain local synchronization if and only if the number of oscillators is less than or equal to $5$. Global synchronization for the individual map of the form $g_c(z)= z^3+ cz$ is also reported.
The purpose of this paper is to address synchronous chaos on the Julia set of complex-valued coupled map lattices (CCMLs). Our main results contain the following. First, we solve an inf min max problem for which its solution gives the fastest synchronized rate in a certain class of coupling matrices. Specifically, we show that for the class of real circulant matrices of size $4$, the coupling weights, possible complex numbers, yielding the fastest synchronized rate can be exactly obtained. Second, for individual map of the form $f_c(z)= z^2+ c$ with $|c|< \frac{1}{4}$, we show that the corresponding CCMLs acquires global synchrony on its Julia set with the number of the oscillators being $3$ or $4$ for the diffusive coupling. For $c=0$ and $-2$, the corresponding CCMLs obtain local synchronization if and only if the number of oscillators is less than or equal to $5$. Global synchronization for the individual map of the form $g_c(z)= z^3+ cz$ is also reported.
2016, 21(1): 185-203
doi: 10.3934/dcdsb.2016.21.185
+[Abstract](1727)
+[PDF](454.0KB)
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The risk of inflation is looming under the current low interest rate environment. Assuming that the investment includes a fixed interest asset and $n$ risky assets under inflation, we consider two scenarios: inflation rate can be observed directly or through a noisy observation. Since the inflation rate is random, all assets become risky. Under this circumstance, we formulate the portfolio selection problem and derive the efficient frontier by solving the associated HJB equation. We find that for a given expected portfolio return, investment at time $t$ is linearly proportional to the price index level. Moreover, the risk for the real value of the portfolio is no longer minimal when all the wealth is put into the fixed interest asset. Finally, for the mutual fund theorem, two funds are needed now instead of the traditional single fund. If an inflation linked bond can be included in the portfolio, the problem is reduced to the traditional mean variance problem with a risk-free and $n+1$ risky assets with real returns.
The risk of inflation is looming under the current low interest rate environment. Assuming that the investment includes a fixed interest asset and $n$ risky assets under inflation, we consider two scenarios: inflation rate can be observed directly or through a noisy observation. Since the inflation rate is random, all assets become risky. Under this circumstance, we formulate the portfolio selection problem and derive the efficient frontier by solving the associated HJB equation. We find that for a given expected portfolio return, investment at time $t$ is linearly proportional to the price index level. Moreover, the risk for the real value of the portfolio is no longer minimal when all the wealth is put into the fixed interest asset. Finally, for the mutual fund theorem, two funds are needed now instead of the traditional single fund. If an inflation linked bond can be included in the portfolio, the problem is reduced to the traditional mean variance problem with a risk-free and $n+1$ risky assets with real returns.
2016, 21(1): 205-225
doi: 10.3934/dcdsb.2016.21.205
+[Abstract](1691)
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In this paper, we consider the long time behavior of the solution for the following nonlinear damped wave equation \begin{eqnarray*} \varepsilon(t) u_{tt}+g(u_{t})-\Delta u+\varphi (u)=f \end{eqnarray*} with Dirichlet boundary condition, in which, the coefficient $\varepsilon$ depends explicitly on time, the damping $g$ is nonlinear and the nonlinearity $\varphi$ has a critical growth. Spirited by this concrete problem, we establish a sufficient and necessary condition for the existence of attractors on time-dependent spaces, which is equivalent to that provided by M. Conti et al.[10]. Furthermore, we give a technical method for verifying compactness of the process via contractive functions. Finally, by the new framework, we obtain the existence of the time-dependent attractors for the wave equations with nonlinear damping.
In this paper, we consider the long time behavior of the solution for the following nonlinear damped wave equation \begin{eqnarray*} \varepsilon(t) u_{tt}+g(u_{t})-\Delta u+\varphi (u)=f \end{eqnarray*} with Dirichlet boundary condition, in which, the coefficient $\varepsilon$ depends explicitly on time, the damping $g$ is nonlinear and the nonlinearity $\varphi$ has a critical growth. Spirited by this concrete problem, we establish a sufficient and necessary condition for the existence of attractors on time-dependent spaces, which is equivalent to that provided by M. Conti et al.[10]. Furthermore, we give a technical method for verifying compactness of the process via contractive functions. Finally, by the new framework, we obtain the existence of the time-dependent attractors for the wave equations with nonlinear damping.
2016, 21(1): 227-243
doi: 10.3934/dcdsb.2016.21.227
+[Abstract](1478)
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We consider gradient systems with an increasing potential that depends on a scalar parameter. As the parameter is varied, critical points of the potential can be eliminated or created through saddle-node bifurcations causing the system to transit from one stable equilibrium located at a (local) minimum point of the potential to another minimum along the heteroclinic connections. These transitions can be represented by a graph. We show that any admissible graph has a realization in the class of two dimensional gradient flows. The relevance of this result is discussed in the context of genesis of hysteresis phenomena. The Preisach hysteresis model is considered as an example.
We consider gradient systems with an increasing potential that depends on a scalar parameter. As the parameter is varied, critical points of the potential can be eliminated or created through saddle-node bifurcations causing the system to transit from one stable equilibrium located at a (local) minimum point of the potential to another minimum along the heteroclinic connections. These transitions can be represented by a graph. We show that any admissible graph has a realization in the class of two dimensional gradient flows. The relevance of this result is discussed in the context of genesis of hysteresis phenomena. The Preisach hysteresis model is considered as an example.
2016, 21(1): 245-252
doi: 10.3934/dcdsb.2016.21.245
+[Abstract](1724)
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In this paper, we study the large-time behavior of solutions to the initial-value problem for the generalized Korteweg--de Vries equation. We show that for initial data in some weighted space, the asymptotic behavior of the solution can be improved. In addition, we give the asymptotic profile of the fundamental solution of the linearized model. We extend and improve the results in [3] and [2].
In this paper, we study the large-time behavior of solutions to the initial-value problem for the generalized Korteweg--de Vries equation. We show that for initial data in some weighted space, the asymptotic behavior of the solution can be improved. In addition, we give the asymptotic profile of the fundamental solution of the linearized model. We extend and improve the results in [3] and [2].
2016, 21(1): 253-269
doi: 10.3934/dcdsb.2016.21.253
+[Abstract](1668)
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In this paper, we investigate the global stability of discrete-time coupled systems with multi-diffusion (DCSMDs). By utilizing a multi-digraph theory, we construct a global Lyapunov function for DCSMDs. Consequently, some sufficient conditions are presented to ensure the stability of a general DCSMDs. Then the proposed theory is successfully applied to analyze the global stability for a discrete-time predator-prey model which is discretized by a nonstandard finite difference scheme. Finally, an example with numerical simulation is given to demonstrate the effectiveness of the obtained results.
In this paper, we investigate the global stability of discrete-time coupled systems with multi-diffusion (DCSMDs). By utilizing a multi-digraph theory, we construct a global Lyapunov function for DCSMDs. Consequently, some sufficient conditions are presented to ensure the stability of a general DCSMDs. Then the proposed theory is successfully applied to analyze the global stability for a discrete-time predator-prey model which is discretized by a nonstandard finite difference scheme. Finally, an example with numerical simulation is given to demonstrate the effectiveness of the obtained results.
2016, 21(1): 271-289
doi: 10.3934/dcdsb.2016.21.271
+[Abstract](1497)
+[PDF](585.7KB)
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The persistence and extinction of an almost periodic epidemic model in a patchy environment are studied. It is shown that the disease cannot invade the disease-free state if the exponential growth bound is less than zero and can invade if it is greater than zero. It is also shown that there exists an almost periodic solution which is globally attractive when each patch admits the same dispersal rate. Finally, numerical simulations illustrate the above results.
The persistence and extinction of an almost periodic epidemic model in a patchy environment are studied. It is shown that the disease cannot invade the disease-free state if the exponential growth bound is less than zero and can invade if it is greater than zero. It is also shown that there exists an almost periodic solution which is globally attractive when each patch admits the same dispersal rate. Finally, numerical simulations illustrate the above results.
2016, 21(1): 291-311
doi: 10.3934/dcdsb.2016.21.291
+[Abstract](1369)
+[PDF](495.4KB)
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An almost periodic epidemic model with age structure in a patchy environment is considered. The existence of the almost periodic disease-free solution and the definition of the basic reproduction ratio $R_{0}$ are given. Based on those, it is shown that a disease dies out if the basic reproduction number $R_{0}$ is less than unity and persists in the population if it is greater than unity.
An almost periodic epidemic model with age structure in a patchy environment is considered. The existence of the almost periodic disease-free solution and the definition of the basic reproduction ratio $R_{0}$ are given. Based on those, it is shown that a disease dies out if the basic reproduction number $R_{0}$ is less than unity and persists in the population if it is greater than unity.
2016, 21(1): 313-335
doi: 10.3934/dcdsb.2016.21.313
+[Abstract](1445)
+[PDF](658.8KB)
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In this paper, we investigate two-vessel gradostat models describing the dynamics of harmful algae with seasonal temperature variations, in which one vessel represents a small cove connected to a larger lake. We first define the basic reproduction number for the model system, and then show that the trivial periodic state is globally asymptotically stable, and algae is washed out eventually if the basic reproduction number is less than unity, while there exists at least one positive periodic state and algal blooms occur when it is greater than unity. There are several types of productions for dissolved toxins, related to the algal growth rate, and nutrient limitation, respectively. For the system with a specific toxin production, the global attractivity of positive periodic steady-state solution can be established. Numerical simulations from the basic reproduction number show that the factor of seasonality plays an important role in the persistence of harmful algae.
In this paper, we investigate two-vessel gradostat models describing the dynamics of harmful algae with seasonal temperature variations, in which one vessel represents a small cove connected to a larger lake. We first define the basic reproduction number for the model system, and then show that the trivial periodic state is globally asymptotically stable, and algae is washed out eventually if the basic reproduction number is less than unity, while there exists at least one positive periodic state and algal blooms occur when it is greater than unity. There are several types of productions for dissolved toxins, related to the algal growth rate, and nutrient limitation, respectively. For the system with a specific toxin production, the global attractivity of positive periodic steady-state solution can be established. Numerical simulations from the basic reproduction number show that the factor of seasonality plays an important role in the persistence of harmful algae.
2016, 21(1): 337-356
doi: 10.3934/dcdsb.2016.21.337
+[Abstract](1479)
+[PDF](495.8KB)
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In this paper, we prove the existence of a $(L_{lu}^2(\mathbb{R}^N)\times L_{lu}^2(\mathbb{R}^N),L_{\rho}^2(\mathbb{R}^N)\times L_{\rho}^2(\mathbb{R}^N))$-global attractor for the solution semigroup generated by the Gray-Scott equations on unbounded domains of space dimension $N\leq3.$
In this paper, we prove the existence of a $(L_{lu}^2(\mathbb{R}^N)\times L_{lu}^2(\mathbb{R}^N),L_{\rho}^2(\mathbb{R}^N)\times L_{\rho}^2(\mathbb{R}^N))$-global attractor for the solution semigroup generated by the Gray-Scott equations on unbounded domains of space dimension $N\leq3.$
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