Monotone dynamical systems: Reflections on new advances &amp;amp; applications

Smith, Hal L.

doi:10.3934/dcds.2017020

Cited by 40 publications

(35 citation statements)

References 88 publications

(142 reference statements)

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“…We start by recalling that a dynamical system is called monotone when the flow generated by this system preserves some partial order 25 . Therefore, the first step is to formally define a partial order induced by a convex cone

K \subset {double-struckR}^{2}

that is typically a quadrant of

R^{2}

for bidimensional dynamical systems 23,31 . Given two elements

{boldz}_{1} = false(x_{1}, y_{1} false) \in K

and

{boldz}_{2} = false(x_{2}, y_{2} false) \in K

, we determine the order relationship as follows:…”

Section: Monotonicity Of the Reduced Modelmentioning

confidence: 99%

A simplified monotone model of Wolbachia invasion encompassing Aedes aegypti mosquitoes

Escobar-Lasso

Vasilieva

2020

Stud Appl Math

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Wolbachia‐based biocontrol has recently emerged as a potential method for the prevention and control of dengue and other vector‐borne diseases. Major vector species, such as Aedes aegypti females, when deliberately infected with Wolbachia become far less capable of getting infected and transmitting the virus to human individuals. In this paper, we propose and qualitatively analyze a simplified model of Wolbachia invasion that describes an interaction and competition between Wolbachia‐infected and wild Aedes aegypti mosquitoes. This model not only facilitates better visualization of the Wolbachia invasion dynamics thanks to its bidimensionality, but it also possesses the noticeable property of monotonicity and exhibits the saddle‐point behavior. The latter helps to identify a threshold manifold consisting of points in R+2 that couple the minimum viable population sizes of wild and Wolbachia‐carrying insects and also regulate the survival and extinction of each population of mosquitoes.

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K \subset {double-struckR}^{2}

that is typically a quadrant of

R^{2}

for bidimensional dynamical systems 23,31 . Given two elements

{boldz}_{1} = false(x_{1}, y_{1} false) \in K

and

{boldz}_{2} = false(x_{2}, y_{2} false) \in K

, we determine the order relationship as follows:…”

Section: Monotonicity Of the Reduced Modelmentioning

confidence: 99%

A simplified monotone model of Wolbachia invasion encompassing Aedes aegypti mosquitoes

Escobar-Lasso

Vasilieva

2020

Stud Appl Math

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“…We introduce a special cone K involving cumulative distributions of the population densities, and a generalized notion of super-and subsolutions of (1.1)-(1.4), where the differential inequalities hold in the sense of the cone K. A comparison principle is then established for the super-and subsolutions, which implies the monotonicity of the semiflow of (1.1)-(1.4) with respect to the cone K (Theorem 2.1). From a theoretical point of view, this paper introduces a new class of reaction-diffusion models with order-preserving property, which may be of independent interest[35].A first application of the monotonicity result yields a simple proof of the existence and global attractivity of the unique positive steady state (or time-periodic solution) to the single species problem (Proposition 3.11). A second application concerns the dynamics of two competing phytoplankton species, as modeled by (1.1)-(1.4), in which sufficient conditions for local (Propositions 4.5 and 4.6) and global stability of semitrivial steady states (Theorems 2.2-2.4) are obtained.…”

mentioning

confidence: 99%

Monotonicity and Global Dynamics of a Nonlocal Two-Species Phytoplankton Model

Jiang¹,

Lam²,

Lou³

et al. 2019

SIAM J. Appl. Math.

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We investigate a nonlocal reaction-diffusion-advection system modeling the population dynamics of two competing phytoplankton species in a eutrophic environment, where nutrients are in abundance and the species are limited by light only for their metabolism. We first demonstrate that the system does not preserve the competitive order in the pointwise sense. Then we introduce a special cone K involving the cumulative distributions of the population densities, and a generalized notion of super-and subsolutions of the nonlocal competition system where the differential inequalities hold in the sense of the cone K. A comparison principle is then established for such super-and subsolutions, which implies the monotonicity of the underlying semiflow with respect to the cone K (Theorem 2.1). As application, we study the global dynamics of the single species system and the competition system. The latter has implications for the evolution of movement for phytoplankton species.

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“…Proof By substituting