Non-constant positive steady states of a prey–predator system with cross-diffusions

In this manuscript, we consider an extended version of the prey–predator system with nonlinear harvesting [Gupta et al., 2015] by introducing a top predator (omnivore) which feeds on more than one trophic levels. Consideration of third species as omnivore makes the system a food web of three populations. We have guaranteed positivity as well as the boundedness of solutions of the proposed system. We observed that the presence of third species complicates the dynamical behavior of the system. It is also observed that multiple positive steady states exist for the proposed system which makes the problem more interesting compared to the similar models studied previously. Sotomayor’s theorem is being utilized to study the saddle-node bifurcation. The persistence conditions are discussed for the proposed model. The local existence of periodic solution through Hopf bifurcations is also guaranteed numerically. It is observed that the proposed model is capable to exhibit more complicated dynamics in the form of chaos in both the cases when there are unique and multiple coexisting steady states. Bifurcation diagrams and Lyapunov exponents have been drawn to ensure the existence of chaotic dynamics of the system.

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Positive steady states of a SI epidemic model with cross diffusion

Applied Mathematics and Computation ◽

10.1016/j.amc.2021.126423 ◽

2021 ◽

Vol 410 ◽

pp. 126423

Author(s):

Nishith Mohan ◽

Nitu Kumari

Keyword(s):

Epidemic Model ◽

Steady States ◽

Cross Diffusion ◽

Positive Steady States

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Zero Eigenvalue Analysis for the Determination of Multiple Steady States in Reaction Networks

Zeitschrift für Naturforschung A ◽

10.1515/zna-1998-3-411 ◽

1998 ◽

Vol 53 (3-4) ◽

pp. 171-177

Author(s):

Hsing-Ya Li

Keyword(s):

Steady State ◽

Rate Constants ◽

Reaction Network ◽

Steady States ◽

Eigenvalue Analysis ◽

Zero Eigenvalue ◽

Positive Steady State ◽

Positive Steady States ◽

Positive Rate ◽

System Of Inequalities

Abstract A chemical reaction network can admit multiple positive steady states if and only if there exists a positive steady state having a zero eigenvalue with its eigenvector in the stoichiometric subspace. A zero eigenvalue analysis is proposed which provides a necessary and sufficient condition to determine the possibility of the existence of such a steady state. The condition forms a system of inequalities and equations. If a set of solutions for the system is found, then the network under study is able to admit multiple positive steady states for some positive rate constants. Otherwise, the network can exhibit at most one steady state, no matter what positive rate constants the system might have. The construction of a zero-eigenvalue positive steady state and a set of positive rate constants is also presented. The analysis is demonstrated by two examples.

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Families of toric chemical reaction networks

Journal of Mathematical Chemistry ◽

10.1007/s10910-020-01162-x ◽

2020 ◽

Vol 58 (9) ◽

pp. 2061-2093

Author(s):

Michael F. Adamer ◽

Martin Helmer

Keyword(s):

Chemical Reaction ◽

Chemical Reaction Networks ◽

Steady States ◽

Reaction Networks ◽

Core Network ◽

Entire Family ◽

Data Set ◽

The Family ◽

Positive Steady States ◽

State Behaviour

Abstract We study families of chemical reaction networks whose positive steady states are toric, and therefore can be parameterized by monomials. Families are constructed algorithmically from a core network; we show that if a family member is multistationary, then so are all subsequent networks in the family. Further, we address the questions of model selection and experimental design for families by investigating the algebraic dependencies of the chemical concentrations using matroids. Given a family with toric steady states and a constant number of conservation relations, we construct a matroid that encodes important information regarding the steady state behaviour of the entire family. Among other things, this gives necessary conditions for the distinguishability of families of reaction networks with respect to a data set of measured chemical concentrations. We illustrate our results using multi-site phosphorylation networks.

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