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1531-3492
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Discrete & Continuous Dynamical Systems - B
September 2016 , Volume 21 , Issue 7
Special issue on analysis of noise and stochastic dynamics in biology
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2016, 21(7): i-ii
doi: 10.3934/dcdsb.201607i
+[Abstract](1440)
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Abstract:
Stochasticity, sometimes referred to as noise, is unavoidable in biological systems. Noise, which exists at all biological scales ranging from gene expressions to ecosystems, can be detrimental or sometimes beneficial by performing unexpected tasks to improve biological functions. Often, the complexity of biological systems is a consequence of dealing with uncertainty and noise, and thus, consideration of noise is necessary in mathematical models. Recent advancement of technology allows experimental measurement on stochastic effects, showing multifaceted and perplexed roles of noise. As interrogating internal or external noise becomes possible experimentally, new models and mathematical theory are needed. Over the past few decades, stochastic analysis and the theory of nonautonomous and random dynamical systems have started to show their strong promise and relevance in studying complex biological systems. This special issue represents a collection of recent advances in this emerging research area.
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Stochasticity, sometimes referred to as noise, is unavoidable in biological systems. Noise, which exists at all biological scales ranging from gene expressions to ecosystems, can be detrimental or sometimes beneficial by performing unexpected tasks to improve biological functions. Often, the complexity of biological systems is a consequence of dealing with uncertainty and noise, and thus, consideration of noise is necessary in mathematical models. Recent advancement of technology allows experimental measurement on stochastic effects, showing multifaceted and perplexed roles of noise. As interrogating internal or external noise becomes possible experimentally, new models and mathematical theory are needed. Over the past few decades, stochastic analysis and the theory of nonautonomous and random dynamical systems have started to show their strong promise and relevance in studying complex biological systems. This special issue represents a collection of recent advances in this emerging research area.
For more information please click the “Full Text” above.
2016, 21(7): 2073-2089
doi: 10.3934/dcdsb.2016037
+[Abstract](2802)
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Abstract:
Environmental variability is often incorporated in a mathematical model by modifying the parameters in the model. In the present investigation, two common methods to incorporate the effects of environmental variability in stochastic differential equation models are studied. The first approach hypothesizes that the parameter satisfies a mean-reverting stochastic process. The second approach hypothesizes that the parameter is a linear function of Gaussian white noise. The two approaches are discussed and compared analytically and computationally. Properties of several mean-reverting processes are compared with respect to nonnegativity and their asymptotic stationary behavior. The effects of different environmental variability assumptions on population size and persistence time for simple population models are studied and compared. Furthermore, environmental data are examined for a gold mining stock. It is concluded that mean-reverting processes possess several advantages over linear functions of white noise in modifying parameters for environmental variability.
Environmental variability is often incorporated in a mathematical model by modifying the parameters in the model. In the present investigation, two common methods to incorporate the effects of environmental variability in stochastic differential equation models are studied. The first approach hypothesizes that the parameter satisfies a mean-reverting stochastic process. The second approach hypothesizes that the parameter is a linear function of Gaussian white noise. The two approaches are discussed and compared analytically and computationally. Properties of several mean-reverting processes are compared with respect to nonnegativity and their asymptotic stationary behavior. The effects of different environmental variability assumptions on population size and persistence time for simple population models are studied and compared. Furthermore, environmental data are examined for a gold mining stock. It is concluded that mean-reverting processes possess several advantages over linear functions of white noise in modifying parameters for environmental variability.
2016, 21(7): 2091-2107
doi: 10.3934/dcdsb.2016038
+[Abstract](2135)
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Abstract:
We study the effect of small real noise on the jump behavior near a singular fold point, which is an important step in understanding the burst-spike behavior in many biological models. We show by the theory of center manifolds and random invariant manifolds that if the order of the noise is high enough, trajectories essentially pass the fold point in the manner as though there is no noise.
We study the effect of small real noise on the jump behavior near a singular fold point, which is an important step in understanding the burst-spike behavior in many biological models. We show by the theory of center manifolds and random invariant manifolds that if the order of the noise is high enough, trajectories essentially pass the fold point in the manner as though there is no noise.
2016, 21(7): 2109-2127
doi: 10.3934/dcdsb.2016039
+[Abstract](2795)
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Abstract:
In this paper, we consider the stochastic vector-host epidemic model with direct transmission. First, we study the existence of a positive global solution and stochastic boundedness of the system of stochastic differential equations which describes the model. Then we introduce the basic reproductive number $\mathcal{R}^s_0$ in the stochastic model, which reflects the deterministic counterpart, and investigate the dynamics of the stochastic epidemic model when $\mathcal{R}^s_0 <1$ and $\mathcal{R}^s_0 >1$. In particular, we show that random effects may lead to extinction in the stochastic case while the deterministic model predicts persistence. Additionally, we provide conditions for the extinction of the infection and stochastic stability of the solution. Finally, numerical simulations are presented to illustrate some of the theoretical results.
In this paper, we consider the stochastic vector-host epidemic model with direct transmission. First, we study the existence of a positive global solution and stochastic boundedness of the system of stochastic differential equations which describes the model. Then we introduce the basic reproductive number $\mathcal{R}^s_0$ in the stochastic model, which reflects the deterministic counterpart, and investigate the dynamics of the stochastic epidemic model when $\mathcal{R}^s_0 <1$ and $\mathcal{R}^s_0 >1$. In particular, we show that random effects may lead to extinction in the stochastic case while the deterministic model predicts persistence. Additionally, we provide conditions for the extinction of the infection and stochastic stability of the solution. Finally, numerical simulations are presented to illustrate some of the theoretical results.
2016, 21(7): 2129-2143
doi: 10.3934/dcdsb.2016040
+[Abstract](2129)
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Abstract:
In this work we study semi-Kolmogorov models for predation with both the carrying capacities and the indirect effects varying with respect to randomly fluctuating environments. In particular, we consider one random semi-Kolmogorov system involving random and essentially bounded parameters, and one stochastic semi-Kolmogorov system involving white noise and stochastic parameters defined upon a continuous-time Markov chain. For both systems we investigate the existence and uniqueness of solutions, as well as positiveness and boundedness of solutions. For the random semi-Kolmogorov system we also obtain sufficient conditions for the existence of a global random attractor.
In this work we study semi-Kolmogorov models for predation with both the carrying capacities and the indirect effects varying with respect to randomly fluctuating environments. In particular, we consider one random semi-Kolmogorov system involving random and essentially bounded parameters, and one stochastic semi-Kolmogorov system involving white noise and stochastic parameters defined upon a continuous-time Markov chain. For both systems we investigate the existence and uniqueness of solutions, as well as positiveness and boundedness of solutions. For the random semi-Kolmogorov system we also obtain sufficient conditions for the existence of a global random attractor.
2016, 21(7): 2145-2168
doi: 10.3934/dcdsb.2016041
+[Abstract](2202)
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Abstract:
Invariance is a crucial property for many mathematical models of biological or biomedical systems, meaning that the solutions necessarily take values in a given range. This property reflects physical or biological constraints of the system and is independent of the model under consideration. While most classical deterministic models respect invariance, many recent stochastic extensions violate this fundamental property. Based on an invariance criterion for systems of stochastic differential equations we discuss several stochastic models exhibiting this behavior and propose classes of modified, admissible models as possible resolutions. Numerical simulations are presented to illustrate the model behavior.
Invariance is a crucial property for many mathematical models of biological or biomedical systems, meaning that the solutions necessarily take values in a given range. This property reflects physical or biological constraints of the system and is independent of the model under consideration. While most classical deterministic models respect invariance, many recent stochastic extensions violate this fundamental property. Based on an invariance criterion for systems of stochastic differential equations we discuss several stochastic models exhibiting this behavior and propose classes of modified, admissible models as possible resolutions. Numerical simulations are presented to illustrate the model behavior.
2016, 21(7): 2169-2191
doi: 10.3934/dcdsb.2016042
+[Abstract](2604)
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Abstract:
While feeding on host plants, viruliferous insects serve as vectors for viruses. Successful viral transmission depends on vector behavior. Two behaviors that impact viral transmission are vector aggregation and dispersal. Vector aggregation may be due to chemical or visual cues or feeding preferences. Vector dispersal can result in widespread disease outbreaks among susceptible host plants. These two behaviors are investigated in plant-vector-virus models. Deterministic and stochastic models are formulated to account for stages of infection, vector aggregation and local dispersal between adjacent crops. First, models for a single crop are studied with aggregation included implicitly through the acquisition and inoculation rates. Second, models with aggregation and dispersal of vectors are studied when one field contains a disease-sensitive crop and another a disease-resistant crop. Analytical expressions are computed for the basic reproduction number in the deterministic models and for the probability of disease extinction in the stochastic models. These two expressions provide useful measures to assess effects of aggregation and dispersal on the rate of disease spread within and between crops and the potential for an outbreak. The modeling framework is based on cassava mosaic virus that causes significant damage in cassava crops in Africa.
While feeding on host plants, viruliferous insects serve as vectors for viruses. Successful viral transmission depends on vector behavior. Two behaviors that impact viral transmission are vector aggregation and dispersal. Vector aggregation may be due to chemical or visual cues or feeding preferences. Vector dispersal can result in widespread disease outbreaks among susceptible host plants. These two behaviors are investigated in plant-vector-virus models. Deterministic and stochastic models are formulated to account for stages of infection, vector aggregation and local dispersal between adjacent crops. First, models for a single crop are studied with aggregation included implicitly through the acquisition and inoculation rates. Second, models with aggregation and dispersal of vectors are studied when one field contains a disease-sensitive crop and another a disease-resistant crop. Analytical expressions are computed for the basic reproduction number in the deterministic models and for the probability of disease extinction in the stochastic models. These two expressions provide useful measures to assess effects of aggregation and dispersal on the rate of disease spread within and between crops and the potential for an outbreak. The modeling framework is based on cassava mosaic virus that causes significant damage in cassava crops in Africa.
2016, 21(7): 2193-2210
doi: 10.3934/dcdsb.2016043
+[Abstract](1943)
+[PDF](913.3KB)
Abstract:
Random fluctuations of an environment are common in ecological and economical settings. We consider a family of concave quadratic polynomials on the unit interval that model a self-limiting growth behavior. The maps are parametrized by an independent, identically distributed random parameter. We show the existence of a unique invariant ergodic measure of the resulting random dynamical system. Moreover, there is an attenuation of the mean of the state variable compared to the constant environment with the averaged parameter.
Random fluctuations of an environment are common in ecological and economical settings. We consider a family of concave quadratic polynomials on the unit interval that model a self-limiting growth behavior. The maps are parametrized by an independent, identically distributed random parameter. We show the existence of a unique invariant ergodic measure of the resulting random dynamical system. Moreover, there is an attenuation of the mean of the state variable compared to the constant environment with the averaged parameter.
2016, 21(7): 2211-2231
doi: 10.3934/dcdsb.2016044
+[Abstract](2260)
+[PDF](2184.9KB)
Abstract:
We study the dynamics of stationary bumps in continuum neural field equations near a saddle-node bifurcation. The integral terms of these evolution equations have a weight kernel describing synaptic interactions between neurons at different locations of the network. Excited regions of the neural field correspond to parts of the domain whose fraction of active neurons exceeds a sharp threshold of a firing rate nonlinearity. For sufficiently low firing threshold, a stable bump coexists with an unstable bump and a homogeneous quiescent state. As the threshold is increased, the stable and unstable branch of bump solutions annihilate in a saddle-node bifurcation. Near this criticality, we derive a quadratic amplitude equation that describes the slow evolution of the even mode (bump contractions) as it depends on the distance from the bifurcation. Beyond the bifurcation, bumps eventually become extinct, and the lifetime of bumps increases for systems nearer the bifurcation. When noise is incorporated, a stochastic amplitude equation for the even mode can be derived, which can be analyzed to quantify bump extinction time both below and above the saddle-node.
We study the dynamics of stationary bumps in continuum neural field equations near a saddle-node bifurcation. The integral terms of these evolution equations have a weight kernel describing synaptic interactions between neurons at different locations of the network. Excited regions of the neural field correspond to parts of the domain whose fraction of active neurons exceeds a sharp threshold of a firing rate nonlinearity. For sufficiently low firing threshold, a stable bump coexists with an unstable bump and a homogeneous quiescent state. As the threshold is increased, the stable and unstable branch of bump solutions annihilate in a saddle-node bifurcation. Near this criticality, we derive a quadratic amplitude equation that describes the slow evolution of the even mode (bump contractions) as it depends on the distance from the bifurcation. Beyond the bifurcation, bumps eventually become extinct, and the lifetime of bumps increases for systems nearer the bifurcation. When noise is incorporated, a stochastic amplitude equation for the even mode can be derived, which can be analyzed to quantify bump extinction time both below and above the saddle-node.
2016, 21(7): 2233-2254
doi: 10.3934/dcdsb.2016045
+[Abstract](1915)
+[PDF](561.0KB)
Abstract:
A nonlocal stochastic model for intra- and extracellular proton dynamics in a tumor is proposed. The intracellular dynamics is governed by an SDE coupled to a reaction-diffusion equation for the extracellular proton concentration on the macroscale. In a more general context the existence and uniqueness of solutions for local and nonlocal SDE-PDE systems are established allowing, in particular, to analyze the proton dynamics model, both in its local version and in the case with nonlocal path dependence. Numerical simulations are performed to illustrate the behavior of solutions, providing some insights into the effects of randomness on tumor acidity.
A nonlocal stochastic model for intra- and extracellular proton dynamics in a tumor is proposed. The intracellular dynamics is governed by an SDE coupled to a reaction-diffusion equation for the extracellular proton concentration on the macroscale. In a more general context the existence and uniqueness of solutions for local and nonlocal SDE-PDE systems are established allowing, in particular, to analyze the proton dynamics model, both in its local version and in the case with nonlocal path dependence. Numerical simulations are performed to illustrate the behavior of solutions, providing some insights into the effects of randomness on tumor acidity.
2016, 21(7): 2255-2273
doi: 10.3934/dcdsb.2016046
+[Abstract](2064)
+[PDF](441.6KB)
Abstract:
In volume transmission, neurons in one brain nucleus send their axons to a second nucleus where neurotransmitter is released into the extracellular space. One would like methods to calculate the average amount of neurotransmitter at different parts of the extracellular space, depending on neural properties and the geometry of the projections and the extracellular space. This question is interesting mathematically because the neuron terminals are both the sources (when they are firing) and the sinks (when they are quiescent) of neurotransmitter. We show how to formulate the questions as boundary value problems for the heat equation with stochastically switching boundary conditions. In one space dimension, we derive explicit formulas for the average concentration in terms of the parameters of the problems in two simple prototype examples and then explain how the same methods can be used to solve the general problem. Applications of the mathematical results to the neuroscience context are discussed.
In volume transmission, neurons in one brain nucleus send their axons to a second nucleus where neurotransmitter is released into the extracellular space. One would like methods to calculate the average amount of neurotransmitter at different parts of the extracellular space, depending on neural properties and the geometry of the projections and the extracellular space. This question is interesting mathematically because the neuron terminals are both the sources (when they are firing) and the sinks (when they are quiescent) of neurotransmitter. We show how to formulate the questions as boundary value problems for the heat equation with stochastically switching boundary conditions. In one space dimension, we derive explicit formulas for the average concentration in terms of the parameters of the problems in two simple prototype examples and then explain how the same methods can be used to solve the general problem. Applications of the mathematical results to the neuroscience context are discussed.
2016, 21(7): 2275-2291
doi: 10.3934/dcdsb.2016047
+[Abstract](2706)
+[PDF](4573.4KB)
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Epithelial-mesenchymal transition (EMT) is an instance of cellular plasticity that plays critical roles in development, regeneration and cancer progression. Recent studies indicate that the transition between epithelial and mesenchymal states is a multi-step and reversible process in which several intermediate phenotypes might coexist. These intermediate states correspond to various forms of stem-like cells in the EMT system, but the function of the multi-step transition or the multiple stem cell phenotypes is unclear. Here, we use mathematical models to show that multiple intermediate phenotypes in the EMT system help to attenuate the overall fluctuations of the cell population in terms of phenotypic compositions, thereby stabilizing a heterogeneous cell population in the EMT spectrum. We found that the ability of the system to attenuate noise on the intermediate states depends on the number of intermediate states, indicating the stem-cell population is more stable when it has more sub-states. Our study reveals a novel advantage of multiple intermediate EMT phenotypes in terms of systems design, and it sheds light on the general design principle of heterogeneous stem cell population.
Epithelial-mesenchymal transition (EMT) is an instance of cellular plasticity that plays critical roles in development, regeneration and cancer progression. Recent studies indicate that the transition between epithelial and mesenchymal states is a multi-step and reversible process in which several intermediate phenotypes might coexist. These intermediate states correspond to various forms of stem-like cells in the EMT system, but the function of the multi-step transition or the multiple stem cell phenotypes is unclear. Here, we use mathematical models to show that multiple intermediate phenotypes in the EMT system help to attenuate the overall fluctuations of the cell population in terms of phenotypic compositions, thereby stabilizing a heterogeneous cell population in the EMT spectrum. We found that the ability of the system to attenuate noise on the intermediate states depends on the number of intermediate states, indicating the stem-cell population is more stable when it has more sub-states. Our study reveals a novel advantage of multiple intermediate EMT phenotypes in terms of systems design, and it sheds light on the general design principle of heterogeneous stem cell population.
2016, 21(7): 2293-2319
doi: 10.3934/dcdsb.2016048
+[Abstract](2484)
+[PDF](566.7KB)
Abstract:
Population systems are often subject to various different types of environmental noises. This paper considers a class of Kolmogorov-type systems perturbed by three different types of noise including Brownian motions, Markovian switching processes, and Poisson jumps, which is described by a regime-switching jump diffusion process. This paper examines these three different types of noises and determines their effects on the properties of the systems. The properties to be studied include existence and uniqueness of global positive solutions, boundedness of this positive solution, and asymptotic growth property, and extinction in the senses of the almost sure and the $p$th moment. Finally, this paper also considers a stochastic Lotka-Volterra system with regime-switching jump diffusion processes as a special case.
Population systems are often subject to various different types of environmental noises. This paper considers a class of Kolmogorov-type systems perturbed by three different types of noise including Brownian motions, Markovian switching processes, and Poisson jumps, which is described by a regime-switching jump diffusion process. This paper examines these three different types of noises and determines their effects on the properties of the systems. The properties to be studied include existence and uniqueness of global positive solutions, boundedness of this positive solution, and asymptotic growth property, and extinction in the senses of the almost sure and the $p$th moment. Finally, this paper also considers a stochastic Lotka-Volterra system with regime-switching jump diffusion processes as a special case.
2016, 21(7): 2321-2336
doi: 10.3934/dcdsb.2016049
+[Abstract](2516)
+[PDF](1734.2KB)
Abstract:
We propose a stochastic logistic model with mate limitation and stochastic immigration. Incorporating stochastic immigration into a continuous time Markov chain model, we derive and analyze the associated master equation. By a standard result, there exists a unique globally stable positive stationary distribution. We show that such stationary distribution admits a bimodal profile which implies that a strong Allee effect exists in the stochastic model. Such strong Allee effect disappears and threshold phenomenon emerges as the total population size goes to infinity. Stochasticity vanishes and the model becomes deterministic as the total population size goes to infinity. This implies that there is only one possible fate (either to die out or survive) for a species constrained to a specific community and whether population eventually goes extinct or persists does not depend on initial population density but on a critical inherent constant determined by birth, death and mate limitation. Such a conclusion interprets differently from the classical ordinary differential equation model and thus a paradox on strong Allee effect occurs. Such paradox illustrates the diffusion theory's dilemma.
We propose a stochastic logistic model with mate limitation and stochastic immigration. Incorporating stochastic immigration into a continuous time Markov chain model, we derive and analyze the associated master equation. By a standard result, there exists a unique globally stable positive stationary distribution. We show that such stationary distribution admits a bimodal profile which implies that a strong Allee effect exists in the stochastic model. Such strong Allee effect disappears and threshold phenomenon emerges as the total population size goes to infinity. Stochasticity vanishes and the model becomes deterministic as the total population size goes to infinity. This implies that there is only one possible fate (either to die out or survive) for a species constrained to a specific community and whether population eventually goes extinct or persists does not depend on initial population density but on a critical inherent constant determined by birth, death and mate limitation. Such a conclusion interprets differently from the classical ordinary differential equation model and thus a paradox on strong Allee effect occurs. Such paradox illustrates the diffusion theory's dilemma.
2016, 21(7): 2337-2361
doi: 10.3934/dcdsb.2016050
+[Abstract](5272)
+[PDF](606.5KB)
Abstract:
This article outlines an attempt to lay the groundwork for understanding stochastic dynamical descriptions of biological processes in terms of a discrete-state space, discrete-time random dynamical system (RDS), or random transformation approach. Such mathematics is not new for continuous systems, but the discrete state space formulation significantly reduces the technical requirements for its introduction to a much broader audiences. In particular, we establish some elementary contradistinctions between Markov chain (MC) and RDS descriptions of a stochastic dynamics. It is shown that a given MC is compatible with many possible RDS, and we study in particular the corresponding RDS with maximum metric entropy. Specifically, we show an emergent behavior of an MC with a unique absorbing and aperiodic communicating class, after all the trajectories of the RDS synchronizes. In biological modeling, it is now widely acknowledged that stochastic dynamics is a more complete description of biological reality than deterministic equations; here we further suggest that the RDS description could be a more refined description of stochastic dynamics than a Markov process. Possible applications of discrete-state RDS are systems with fluctuating law of motion, or environment, rather than inherent stochastic movements of individuals.
This article outlines an attempt to lay the groundwork for understanding stochastic dynamical descriptions of biological processes in terms of a discrete-state space, discrete-time random dynamical system (RDS), or random transformation approach. Such mathematics is not new for continuous systems, but the discrete state space formulation significantly reduces the technical requirements for its introduction to a much broader audiences. In particular, we establish some elementary contradistinctions between Markov chain (MC) and RDS descriptions of a stochastic dynamics. It is shown that a given MC is compatible with many possible RDS, and we study in particular the corresponding RDS with maximum metric entropy. Specifically, we show an emergent behavior of an MC with a unique absorbing and aperiodic communicating class, after all the trajectories of the RDS synchronizes. In biological modeling, it is now widely acknowledged that stochastic dynamics is a more complete description of biological reality than deterministic equations; here we further suggest that the RDS description could be a more refined description of stochastic dynamics than a Markov process. Possible applications of discrete-state RDS are systems with fluctuating law of motion, or environment, rather than inherent stochastic movements of individuals.
2016, 21(7): 2363-2378
doi: 10.3934/dcdsb.2016051
+[Abstract](2652)
+[PDF](473.1KB)
Abstract:
We study stochastic versions of a deterministic SIRS(Susceptible, Infective, Recovered, Susceptible) epidemic model with standard incidence. We study the existence of a stationary distribution of stochastic system by the theory of integral Markov semigroup. We prove the distribution densities of the solutions can converge to an invariant density in $L^1$. This shows the system is ergodic. The presented results are demonstrated by numerical simulations.
We study stochastic versions of a deterministic SIRS(Susceptible, Infective, Recovered, Susceptible) epidemic model with standard incidence. We study the existence of a stationary distribution of stochastic system by the theory of integral Markov semigroup. We prove the distribution densities of the solutions can converge to an invariant density in $L^1$. This shows the system is ergodic. The presented results are demonstrated by numerical simulations.
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