Commensal Symbiosis Research Articles

Stability and bifurcation in a two-patch commensal symbiosis model with nonlinear dispersal and additive Allee effect

In this paper, a two-patch model with additive Allee effect, nonlinear dispersal and commensalism is proposed and studied. The stability of equilibria and the existence of saddle-node bifurcation, transcritical bifurcation are discussed. Through qualitative analysis of the model, we know that the persistence and the extinction of population are influenced by the Allee effect, dispersal and commensalism. Combining with numerical simulation, the study shows that the total population density will increase when the Allee effect constant [Formula: see text] increases or [Formula: see text] decreases. In addition to suppress the Allee effect, nonlinear dispersal and commensalism are crucial to the survival of the species in the two patches.

The role of industrial symbiosis on consumer demand under ecological carbon fixation: Evidence from the energy industry

AbstractSince the energy sector is the primary source of carbon emissions, achieving low carbon emission reduction and meeting the carbon peak and carbon neutrality targets require a full grasp of the symbiotic relationship between the traditional and new energy industries. This study analyzes data from China's conventional energy industries and the new energy industry from 2005 to 2021 to examine the symbiosis between these two sectors of the economy and its implications. The Lotka–Volterra model is used to determine the symbiosis between industries and further select provinces with industrial symbiosis to analyze the relationship between industrial symbiosis and consumer demand. The results show that, first, 28 provinces in China have industrial symbiosis, of which only four provinces are commensal symbiosis and the remaining 24 provinces are parasitic symbiosis. Second, whereas the parasitic symbiosis will impede the growth of consumer demand, the commensal symbiosis between the new energy industry and the traditional energy industry can greatly encourage the growth of consumer demand. In addition, the promotion effect of commensal symbiosis on consumer demand is mainly manifested in government consumption, while parasitic symbiosis has a dampening effect on residential consumption, government consumption, and investment. Finally, under the regulation of ecological carbon sequestration, parasitic symbiosis can promote the growth of consumer demand. Therefore, the government should continue to promote industrial symbiosis and ecological carbon sequestration and, in this way, promote the growth of consumer demand.

Dynamics of stability, bifurcation and control for a commensal symbiosis model

Dynamics of stability, bifurcation and control for a commensal symbiosis model

Bifurcation Behavior and Hybrid Controller Design of a 2D Lotka–Volterra Commensal Symbiosis System Accompanying Delay

All the time, differential dynamical models with delay has witness a tremendous application value in characterizing the internal law among diverse biological populations in biology. In the current article, on the basis of the previous publications, we formulate a new Lotka–Volterra commensal symbiosis system accompanying delay. Utilizing fixed point theorem, inequality tactics and an appropriate function, we gain the sufficient criteria on existence and uniqueness, non-negativeness and boundedness of the solution to the formulated delayed Lotka–Volterra commensal symbiosis system. Making use of stability and bifurcation theory of delayed differential equation, we focus on the emergence of bifurcation behavior and stability nature of the formulated delayed Lotka–Volterra commensal symbiosis system. A new delay-independent stability and bifurcation conditions on the model are presented. By constructing a positive definite function, we explore the global stability. By constructing two diverse hybrid delayed feedback controllers, we can adjusted the domain of stability and time of appearance of Hopf bifurcation of the delayed Lotka–Volterra commensal symbiosis system. The effect of time delay on the domain of stability and time of appearance of Hopf bifurcation of the model is given. Matlab experiment diagrams are provided to sustain the acquired key outcomes.

Merdan-type Allee Effect on a Lotka-Volterra Commensal Symbiosis Model with Density-dependent Birth Rate

A Lotka-Volterra commensal symbiosis model with a density dependent birth rate and a Merdan-type Allee effect on the second species has been proposed and examined. The global attractivity of system’s equilibria is ensured by using the differential inequality theory. Our results show that the Allee effect has no effect on the existence or stability of the system’s equilibrium point. However, both species take longer to approach extinction or a stable equilibrium state as the Allee effect increases.

Positive periodic solution of a discrete commensal symbiosis model with Beddington-DeAngelis functional response

A non-autonomous discrete commensal symbiosis model with Beddington-DeAngelis functional response is proposed and studied in this paper. Sufficient conditions are obtained for the existence of positive periodic solution of the system.

Positive Periodic Solution of a Discrete Lotka-volterra Commensal Symbiosis Model with Michaelis-menten Type Harvesting

A non-autonomous discrete Lotka-volterra commensal symbiosis model with Michaelis-Menten type harvesting is proposed and studied in this paper. Under some very simple and easily verified condition, we show that the system admits at least one positive periodic solution.

Dynamic Behaviors of an Obligate Commensal Symbiosis Model with Crowley–Martin Functional Responses

A two species obligate commensal symbiosis model with Crowley–Martin functional response was proposed and studied in this paper. For an autonomous case, local and global dynamic behaviors of the system were investigated, respectively. The conditions that ensure the existence of the positive equilibrium is coincidentla to the conditions of global stability of a positive equilibrium. For nonautonomous cases, persistent and extinction properties of the system are investigated.

Global Stability of a Commensal Symbiosis Model With Holling Ii Functional Response and Feedback Controls

A commensal symbiosis model with Holling II functional response and feedback controls is proposed and studied in this paper. The system admits four equilibria, and three boundary equilibria are unstable, only positive equilibrium is locally asymptotically stable. By applying the comparison theorem of differential equation, we show that the unique positive equilibrium is globally attractive. Numeric simulations show the feasibility of the main result.

Stability and Period-Doubling Bifurcation in a Modified Commensal Symbiosis Model with Allee Effect

In this article, the qualitative behaviour of discrete-time commensal symbiosis model which is obtained by implementing the forward Euler’s scheme is discussed in detail. Firstly, the local stability conditions of fixed points of the model are studied. It is proved that the considered model undergoes Period-Doubling bifurcation around coexistence fixed point with the help of bifurcation theory. In order to support the accuracy of obtained analytical finding, some parameter values have been determined and numerical simulations are carried out for these parameter values. Numerical simulations display new and rich nonlinear dynamical behaviours. More specifically, when the parameter 𝛿 is choosen as a bifurcation parameter, it is seen that the considered discrete-time commensal symbiosis model shows very rich nonlinear dynamical.

Periodic solution of a discrete commensal symbiosis model with Hassell-Varley type functional response

Abstract A non-autonomous discrete commensal symbiosis model with Hassell-Varley type functional response is proposed and studied in this paper. Sufficient conditions are obtained for the existence of positive periodic solution of the system.

Almost periodic solutions of a commensalism system with Michaelis-Menten type harvesting on time scales

Abstract In this paper, we consider an almost periodic commensal symbiosis model with nonlinear harvesting on time scales. We establish a criterion for the existence and uniformly asymptotic stability of unique positive almost periodic solution of the system. Our results show that the continuous system and discrete system can be unify well. Examples and their numerical simulations are carried out to illustrate the feasibility of our main results.

The influence of commensalism on a Lotka\u2013Volterra commensal symbiosis model with Michaelis\u2013Menten type harvesting

In this paper, we study the following Lotka–Volterra commensal symbiosis model of two populations with Michaelis–Menten type harvesting for the first species: \t\t\tdxdt=r1x(1−xK1+αyK1)−qExm1E+m2x,dydt=r2y(1−yK2),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\begin{gathered} \\frac{dx}{dt} = r_{1}x \\biggl(1- \\frac{x}{K_{1}}+\\alpha \\frac{y}{K_{1}} \\biggr)- \\frac{qEx}{m_{1}E+m_{2}x}, \\\\ \\frac{dy}{dt} = r_{2}y \\biggl(1- \\frac{y}{K_{2}} \\biggr), \\end{gathered} $$\\end{document} where r_{1}, r_{2}, K_{1}, K_{2}, α, q, E, m_{1} and m_{2} are all positive constants. The local and global dynamic behaviors of the system are investigated, respectively. For the limited harvesting case (i.e., q is small enough), we show that the system admits a unique globally stable positive equilibrium. For the over harvesting case, if the cooperate intensity of the both species (α) and the capacity of the second species (K_{2}) are large enough, the two species could coexist in a stable state; otherwise, the first species will be driven to extinction. Numeric simulations are carried out to show the feasibility of the main results.

Dynamic behaviors of a Holling type commensal symbiosis model with the first species subject to Allee effect

A two species commensal symbiosis model with Holling type functional response and the first species subject to Allee effect takes the form $$ \di\frac{dx}{dt}&=&x\Big(a_1-b_1x\Big)\di\frac{x}{\beta +x}+\di\frac{c_1xy^p}{1+y^p}, \di\frac{dy}{dt}&=&y(a_2-b_2y) $$ is investigated, where $a_i, b_i, i=1,2$ $p$, $\beta$ and $c_1$ are all positive constants, $p\geq 1$ is a positive integer. Local and global stability property of the equilibria are investigated. Our study indicates that the unique positive equilibrium is globally stable. Also, the final density of the first species is increasing with the Allee effect, this result differs from that obtained for the Holling type commensal symbiosis model with the second species subject to Allee effect.

On the existence and stability of positive periodic solution of a nonautonomous commensal symbiosis model with Michaelis-Menten type harvesting

A non-autonomous commensal symbiosis model of two populations with Michaelis-Menten type harvesting is proposed and studied in this paper. By using a continuation theorem based on Gaines and Mawhin's coincidence degree, we study the global existence of positive periodic solutions of the system. By constructing a suitable Lyapuonov function, sufficient conditions which ensure the global attractivity of the positive periodic solution are obtained. Numeric simulations are carried out to show the feasibility of the main results.

Dynamic behaviors of a commensal symbiosis model involving Allee effect and one party can not survive independently

A two-species commensal symbiosis model involving Allee effect and one party can not survive independently is proposed and studied in this paper. Sufficient conditions which ensure the local and global stability of the boundary equilibrium and the positive equilibrium are obtained, respectively. Numeric simulations show that with the increasing of Alee effect, the system takes much longer time to reach its stable steady-state solution, though the Allee effect has no influence on the final density of the species. The Allee effect has instable effect on the system, however, such effect is controllable.

Allee effect increasing the final density of the species subject to the Allee effect in a Lotka\u2013Volterra commensal symbiosis model

A Lotka–Volterra commensal symbiosis model with first species subject to the Allee effect is proposed and studied in this paper. Local and global stability property of the equilibria are investigated. An amazing finding is that with increasing Allee effect, the final density of the species subject to the Allee effect is also increased. Such a phenomenon is different from the known results, and it is the first time to be observed. Numeric simulations are carried out to show the feasibility of the main results.

A Holling type commensal symbiosis model involving Allee effect

A two species commensal symbiosis model with Holling type functional response and Allee effect on the second species takes the form $$ \di\frac{dx}{dt}&=&x\Big(a_1-b_1x+\di\frac{c_1y^p}{1+y^p}\Big), \di\frac{dy}{dt}&=&y(a_2-b_2y)\di\frac{y}{u+y} $$ is investigated, where $a_i, b_i, i=1,2$ $p$, $u$ and $c_1$ are all positive constants, $p\geq 1$. Local and global stability property of the equilibria is investigated.Our study indicates that the unique positive equilibrium is globally stable and the system always permanent, consequently, Allee effect has no influence on the final density of the species. However, numeric simulations show that the stronger the Allee effect, the longer the for the system to reach its stable steady-state solution.

Dynamic behaviors of a commensal symbiosis model with non-monotonic functional response and non-selective harvesting in a partial closure

A two species commensal symbiosis model with non-monotonic functional response and non-selective harvesting in a partial closure takes the form $$ \di\frac{dx}{dt}&=&x\Big(a_1-b_1x+\di\frac{c_1y }{d_1+y^2}\Big)-q_1Emx, \di\frac{dy}{dt}&=&y(a_2-b_2y)-q_2Emy $$ is proposed and studied, where $a_i, b_i, q_i, i=1,2$ $c_1$, $E$, $m(0<m<1)$ and $d_1$ are all positive constants. Depending on the range of the parameter $m$, the system may be collapse, or partial survival, or the two species could be coexist in a stable state. We also show that if the system admits a unique positive equilibrium, then it is globally asymptotically stable. By introducing the harvesting term and the reserve area, the system exhibit rich dynamic behaviors. Our results generalize the main results of Chen and Wu (A commensal symbiosis model with non-monotonic functional response, Commun. Math. Biol. Neurosci. 2017 (2017), Article ID 5).

A commensal symbiosis model with non-monotonic functional response

A two species commensal symbiosis model with non-monotonic functional response takes the form $$ \di\frac{dx}{dt}=x\Big(a_1-b_1x+\di\frac{c_1y }{d_1+y^2}\Big), \di\frac{dy}{dt}=y(a_2-b_2y) $$ is proposed and studied, where $a_i, b_i, i=1,2$ $c_1$ and $d_1$ are all positive constants. We show that the system admits a unique globally asymptotically stable positive equilibrium. https://doi.org/10.28919/cmbn/2839