Monotone Dynamical Systems Research Articles

Reprogramming multistable monotone systems with application to cell fate control.

Multistability is a key property of dynamical systems modeling cellular regulatory networks implicated in cell fate decisions, where, different stable steady states usually represent distinct cell phenotypes. Monotone network motifs are highly represented in these regulatory networks. In this paper, we leverage the properties of monotone dynamical systems to provide theoretical results that guide the selection of inputs that trigger a transition, i.e., reprogram the network, to a desired stable steady state. We first show that monotone dynamical systems with bounded trajectories admit a minimum and a maximum stable steady state. Then, we provide input choices that are guaranteed to reprogram the system to these extreme steady states. For intermediate states, we provide an input space that is guaranteed to contain an input that reprograms the system to the desired state. We then provide implementation guidelines for finite-time procedures that search this space for such an input, along with rules to prune parts of the space during search. We demonstrate these results on simulations of two recurrent regulatory network motifs: self-activation within mutual antagonism and self-activation within mutual cooperation. Our results depend uniquely on the structure of the network and are independent of specific parameter values.

Global directed dynamics of a Lotka–Volterra competition-diffusion system

In this study, we consider a directed–diffusion system describing the interactions between two organisms in heterogeneous environment. We first establish a linearly stability of the co-existence (positive) steady state. Then we further present a classification on all possible long-time dynamical behaviors by appealing to the theory of monotone dynamical systems.

Existence and Stability of Stationary States of a Reaction–Diffusion-Advection Model for Two Competing Species

It is well known that the research of two species in the Lotka–Volterra competition system could create very interesting dynamics. In our paper, we investigate the global dynamical behavior of a classic Lotka–Volterra competition system by studying the steady states and corresponding stability by mainly employing the methods of monotone dynamical systems theory, Lyapunov–Schmidt reduction and spectral theory and so on. It illustrates that the dynamical behavior substantially relies on certain variable of the maximal growth rate. Furthermore, we obtain that one of the semi-trivial steady state solutions is a global attractor in some special cases. In biology, these results show that both of the species do not coexist and the mutant forces the extinction of resident species under some condition for two similar species system.

Adaptive evolution of life history strategies related to maturation time in seasonal environment

Maturation time plays an important role on life history evolution because of its great influence on reproduction and body size. To study the optimal life history strategy determined by maturation time in detail, a competitive mathematical model with two populations in a seasonal environment is established and analyzed. Firstly, the dynamic behaviors of the population model are investigated by stability analysis and using the theory of monotone dynamical systems. There are four possible dynamic outcomes of the competition model namely extinction, competitive exclusion, competitive coexistence and bistability. Further, the influences of seasonality on the competition results are discussed. Secondly, the evolution of maturation time is studied via adaptive dynamics theory. Evolutionarily stable strategy (ESS) of maturation time is shown to be local and also convergent stable strategy (CSS). And the results reveal that the ESS is different from the strategy of maximum fitness of invasive population. Finally, by using numerical simulations, it is shown that seasonality has an important influence on the mature strategy. The ESS will decrease with production season length elongating along the direction of growth rate declining, and it would not cross the maximum fitness strategy.

Analysis of a reaction-diffusion cholera epidemic model in a spatially heterogeneous environment

In this paper, we derive and analyze a reaction-diffusion cholera model in bounded spatial domain with zero-flux boundary condition and general nonlinear incidence functions. The parameters in the model are space-dependent due to the spatial heterogeneity. By applying the theory of monotone dynamical systems and uniform persistence, we prove that the model admits the global threshold dynamics in terms of the basic reproduction number ℜ0, which is defined by the spectral radius of the next generation operator. When all model parameters are strictly positive constants, we study three types of nonlinear incidence functions to achieve the global stability results on the unique positive cholera-endemic steady state (CESS) whenever it exists. For all these examples, the sharp threshold property based on the basic reproduction number was completely established by using Lyapunov functional techniques under some realistic assumptions. Our numerical results reveal that when ℜ0 > 1, the convergence speed of the solution to the CESS becomes faster as the diffusion coefficient d becomes larger in the spatially homogeneous case. While in the spatially heterogeneous case, cholera can not be controlled by limiting the movement of host individuals, and the spatial heterogeneity does not always enhance the disease persistence.

Population dynamics and evolution in river ecosystems

In this paper, we consider a single species model and a two-species competition model arising in river ecology. An interesting feature of these models lies in the boundary conditions, where there may appear a net loss of individuals from the habitat ends, as measured by two parameters bu and bd. We first determine various sufficient conditions for the persistence of a single species, in terms of the critical habitat size and critical advection rate. Then we turn to study a competition system, where two populations are supposed to differ only in their advection speeds. By using the theory of monotone dynamical systems and principal spectrum, we achieve a complete understanding on the global dynamics, which suggests that weaker advective movement is more beneficial for species to win the competition, extending several existing work (Lou et al., 2016; Xu and Gan, 2019; Zhou and Zhao, 2018).

Bi-stable dynamics of a host-pathogen model

Crop host-pathogen interaction have been a main issue for decades, in particular for food security. In this paper, we focus on the modeling and long term behavior of soil-borne pathogens. We first develop a new compartmental temporal model, which exhibits bi-stable asymptotical dynamics. To investigate the long term behavior, we use LaSalle Invariance Principle to derive sufficient conditions for global asymptotic stability of the pathogen free equilibrium and monotone dynamical systems theory to provide sufficient conditions for permanence of the system. Finally, we develop a partially degenerate reaction diffusion system, providing a numerical exploration based on the results obtained for the temporal system. We show that a traveling wave solution may exist where the speed of the wave follows a power law.

Global stabilization of a genetic positive feedback loop via the design of a synthetic auto-repression

Genetic positive feedback loops are essential for cell differentiation processes. They are accurately modeled with N-dimensional non-linear monotone dynamical systems that display bi-stability: the two stable fixed points represent two distinct cell differentiated states, whereas the unstable fixed point is interpreted as a cell undifferentiated state. This paper shows that the synthetic design of a simple self-inhibition of one gene in the loop is able to globally stabilize the unstable fixed point of the network. This modification may lead to a promising cell dedifferentiation process during which cells regress from a specialized state to an earlier developmental state. Compared to a similar experiment designed for the Toggle Switch, this new synthetic circuit prevents the use of any input and measurement devices, reducing greatly the complexity of the biological set-up. In order to take into account inherent biological uncertainties, the cell undifferentiated state is later considered as a region of the state space around the unstable fixed point and is shown to be globally attractive with the same simple synthetic modification of the loop. Some conditions are given such that all the possible fixed points of the circuit are confined in the undifferentiated region and the global results are proved with the theory of monotone dynamical systems.

Reprogramming cooperative monotone dynamical systems.

Multistable dynamical systems are ubiquitous in nature, especially in the context of regulatory networks controlling cell fate decisions, wherein stable steady states correspond to different cell phenotypes. In the past decade, it has become experimentally possible to "reprogram" the fate of a cell by suitable externally imposed input stimulations. In several of these reprogramming instances, the underlying regulatory network has a known structure and often it falls in the class of cooperative monotone dynamical systems. In this paper, we therefore leverage this structure to provide concrete guidance on the choice of inputs that reprogram a cooperative dynamical system to a desired target steady state. Our results are parameter-independent and therefore can serve as a practical guidance to cell-fate reprogramming experiments.

On the use of the sterile insect release technique to reduce or eliminate mosquito populations

• A bistable monotone dynamical system is used to model mosquito dynamics. • Necessary conditions are obtained to reach elimination for constant, continuous or periodic releases of sterile males. • Application of our results based on a field experiment done in Tetiaroa atoll, French Polynesia. Vector control is critical to limit the circulation of vector-borne diseases, like chikungunya, dengue or zika, which have become important issues around the world. Among them, the Sterile Insect Technique (SIT) and the Incompatible Insect Technique (IIT) have recently aroused a renewed interest . In this paper we derive and study a minimalistic mathematical model designed for Aedes mosquito population elimination by SIT/IIT. Contrary to most of the previous models, it is bistable in general, allowing simultaneously for elimination of the population and for its survival. We consider different types of releases (constant, periodic or impulsive) and show necessary conditions to reach elimination in each case. We also estimate both sufficient and minimal treatment times. Biological parameters are estimated from a case study of an Aedes polynesiensis population, for which extensive numerical investigations illustrate the analytical results. The applications of this work are two-fold: to help identifying some key parameters that may need further field investigations, and to help designing release protocols.

On a vector-host epidemic model with spatial structure

In this paper, we study a reaction–diffusion vector-host epidemic model. We define the basic reproduction number R0 and show that R0 is a threshold parameter: if the disease free equilibrium is globally stable; if R0 > 1 the model has a unique globally stable positive equilibrium. Our proof combines arguments from monotone dynamical system theory, persistence theory, and the theory of asymptotically autonomous semiflows.

Dynamics of a diffusive competition model in spatially heterogeneous environment

We discuss the effects of dispersal rate and spatial heterogeneity on population dynamics via a diffusive Lotka–Volterra competition model under homogeneous Dirichlet boundary conditions. By the theory of monotone dynamical system, we show that in the weak competition case, for some ranges of dispersal rate of competing species 1, if dispersal rate of competing species 2 is sufficiently small, there exists a unique coexistence steady state of the model which is globally asymptotically stable. Moreover, for certain ranges of dispersal rate of competing species 1, we derive the limit behavior of the coexistence steady state as dispersal rate of competing species 2 tends to zero. On the other hand, we demonstrate that for some larger scopes of dispersal rate of competing species 1, if dispersal rate of competing species 2 is small enough, there is no coexistence steady state, whereas one of the semi-trivial steady states of the model is globally asymptotically stable.

Travelling Waves for a Nonlocal Delay Differential Equation

A class of nonlocal delay differential equation with nonmonotone nonlinearity that arises from population dynamics is considered. The existence of travelling waves is established by constructing two auxiliary monotone dynamical systems combined with a global coupling condition. Our results require neither the compactness nor the reflection invariance of the solution semiflow.

Monotone dynamical systems with dense periodic points

In this paper we prove a recent conjecture by M. Hirsch, which says that if (f,Ω) is a discrete time monotone dynamical system, with f:Ω→Ω a homeomorphism on an open connected subset of a finite dimensional vector space, and the periodic points of f are dense in Ω, then f is periodic.

Global dynamics of a Lotka–Volterra competition–diffusion–advection system in heterogeneous environments

We study a Lotka–Volterra type reaction–diffusion–advection system, which describes the competition for the same resources between two aquatic species undergoing different dispersal strategies, as reflected by their diffusion and/or advection rates. For the non-advective case, this problem was solved by Dockery et al. [9], and recently He and Ni [14] provided a further classification on the global dynamics for a more general model. However, the key ideas developed in [9,14] do not appear to work when advection terms are involved. By assuming the resource function is decreasing in the spatial variable, we establish the non-existence of co-existence steady state and perform sufficient analysis on the local stability of two semi-trivial steady states, where new techniques were introduced to overcome the difficulty caused by the non-analyticity of stationary solutions as well as the diffusion–advection type operators. Combining these two aspects with the theory of monotone dynamical systems, we finally obtain the global dynamics, which suggests that the competitive exclusion principle holds in most situations.

Bistable waves of a recursive system arising from seasonal age-structured population models

This paper is devoted to the existence, uniqueness and stability of bistable traveling waves for a recursive system, which is defined by the iterations of the Ponicare map of a yearly periodic age-structured population model derived in the companion paper [ 8 ]. The existence of the wave is established by appealing to a monotone dynamical system theory, and the uniqueness and stability are obtained by employing a squeezing method.

Global dynamics of a classical Lotka–Volterra competition–diffusion–advection system

In this paper, we study a classical two species Lotka–Volterra competition–diffusion–advection system, where the diffusion and advection rates of two competitors are supposed to be proportional. By employing the principal spectral theory, we first establish a key a priori estimate on the co-existence (positive) steady state, which is a powerful tool to link the local and global dynamics. We then further present a complete classification on all possible long-time dynamical behaviors by appealing to the theory of monotone dynamical systems. Lastly, we apply these results to a special situation where two species are competing for the same resources and obtain a sharp criteria in term of certain variable parameters for all kinds of global dynamics. This work gives a positive answer to the conjecture proposed by Lou et al. in [34] by considering a more general model under certain conditions, and also, can be seen as a further development of He and Ni [19] for competition–diffusion system, where we bring new ingredients in the arguments to overcome the difficulty caused by the involvement of advection.

On crop vector-borne diseases. Impact of virus lifespan and contact rate on the traveling-wave speed of infective fronts

Diseases, in particular, vector-borne diseases are very important issues in crop protection. However, despite their impact on food safety, very few mathematical models have been developed in order to improve control strategies. Motivated by existing literature, we begin by considering a temporal model of vector-borne diseases in (annual) crops. Using appropriate mathematical methods, we show the existence of threshold parameters and discuss several control strategies based on the model. Then, the model is modified to include a spatial component in order to take into account that vectors are moving. We study both theoretically and numerically the related system and other subsystems that are easier to handle using the theory of monotone dynamical systems. We show that traveling wave solutions may exist, with traveling wave speed dependent on the virus lifespan and the contact rate between the pest and the crop. Finally, we discuss the consequences in terms of control strategies.

A deterministic mathematical model for bidirectional excluded flow with Langmuir kinetics.

In many important cellular processes, including mRNA translation, gene transcription, phosphotransfer, and intracellular transport, biological “particles” move along some kind of “tracks”. The motion of these particles can be modeled as a one-dimensional movement along an ordered sequence of sites. The biological particles (e.g., ribosomes or RNAPs) have volume and cannot surpass one another. In some cases, there is a preferred direction of movement along the track, but in general the movement may be bidirectional, and furthermore the particles may attach or detach from various regions along the tracks. We derive a new deterministic mathematical model for such transport phenomena that may be interpreted as a dynamic mean-field approximation of an important model from mechanical statistics called the asymmetric simple exclusion process (ASEP) with Langmuir kinetics. Using tools from the theory of monotone dynamical systems and contraction theory we show that the model admits a unique steady-state, and that every solution converges to this steady-state. Furthermore, we show that the model entrains (or phase locks) to periodic excitations in any of its forward, backward, attachment, or detachment rates. We demonstrate an application of this phenomenological transport model for analyzing ribosome drop off in mRNA translation.

Stability of Dirac concentrations in an integro-PDE model for evolution of dispersal

We consider an integro-PDE model from evolutionary biology. The solution \(u_\epsilon (x,\alpha )\) is structured by two variables \(x\in D \subset {\mathbb {R}}^k\) and \(\alpha \in (\underline{\alpha },{\overline{\alpha }})\subset \subset {\mathbb {R}}_+\). The diffusion coefficient in the x direction depends on \(\alpha \) and the diffusion coefficient in the \(\alpha \) direction is a constant \(\epsilon ^2\). A special feature of this model is the appearance of the integral \({\hat{u}}_\epsilon (x)\) of the solution in the \(\alpha \) variable, which can be viewed as an infinite dimensional parameter of the problem. In a previous work, the existence of a steady state that exhibits Dirac-concentration in one of the variables yet remains regular in the other variables was proved independently by Lam and Lou (J Funct Anal 272:1755–1790, 2017) and by Perthame and Souganidis (Math Model Nat Phenom 11:154–166, 2016). In this paper, we tackle the long-time dynamics of solutions. When the environment function is non-constant, we show that the steady state is linearly stable by considering the corresponding nonlocal eigenvalue problem. Uniqueness of steady state is obtained from the stability result via a degree argument. When the environment function is a constant, the global asymptotic stability result is obtained. This problem can be regarded as a competition of infinitely many species parameterized by \(\alpha \). As with the competition model for three or more species, the integro-PDE model does not generate a monotone dynamical system so that it is necessary to consider all (real or complex) eigenvalues in determining its linear stability.