The necessary and sufficient conditions for the existence of periodic orbits in a Lotka–Volterra system

AbstractBy extending Darboux method to three dimension, we present necessary and sufficient conditions for the existence of periodic orbits in three species Lotka–Volterra systems with the same intrinsic growth rates. Therefore, all the published sufficient or necessary conditions for the existence of periodic orbits of the system are included in our results. Furthermore, we prove the stability of periodic orbits. Hopf bifurcation is shown for the emergence of periodic orbits and new phenomenon is presented: at critical values, each equilibrium are surrounded by either equilibria or periodic orbits

Dynamics of a rate equation describing cluster-size evolution

AbstractIn this paper, we show dynamics of Smoluchowski's rate equation which has been widely applied to studies of aggregation processes (i.e., the evolution of cluster-size distribution) in physics. We introduce dissociation in the rate equation while dissociation is neglected in previous works. We prove the positiveness of solutions of the equation, which is a basic guarantee for the effectiveness of the model since the possibility that some solution may be negative is excluded. For the case of cluster coalesce without dissociation, we show both the equilibrium uniqueness and the equilibrium stability under the condition that the monomer deposition stops. For the case that clusters evolve with dissociation and there is no monomer deposition, we show the equilibrium uniqueness and prove the equilibrium stability if the maximum cluster size is not larger than three while we show the equilibrium stability by numerical simulations if the maximum size is larger than three

Agent-Based Load Balancing on Homogeneous Minigrids: Macroscopic Modeling and Characterization

Abstract—In this paper, we present a macroscopic characterization of agent-based load balancing in homogeneous minigrid environments. The agent-based load balancing is regarded as agent distribution from a macroscopic point of view. We study two quantities on minigrids: the number and size of teams where agents (tasks) queue. In macroscopic modeling, the load balancing mechanism is characterized using differential equations. We show that the load balancing we concern always converges to a steady state. Furthermore, we show that load balancing with different initial distributions converges to the same steady state gradually. Also, we prove that the steady state becomes an even distribution if and only if agents have complete knowledge about agent teams on minigrids. Utility gains and efficiency are introduced to measure the quality of load balancing. Through numerical simulations, we discuss the utility gains and efficiency of load balancing in different cases and give a series of analysis. In order to maximize the utility gain and the efficiency, we theoretically discuss the optimization of agents ’ strategies. Finally, in order to validate our proposed agentbased load balancing mechanism, we develop a computing platform, called Simulation System for Grid Task Distribution (SSGTD). Through experimentation, we note that our experimental results in general confirm our theoretical proofs and numerical simulation results on the proposed equation system. In addition, we find a very interesting phenomenon, that is, our agent-based load balancing mechanism is topology-independent

Global dynamics of a mutualism–competition model with one resource and multiple consumers

Recent simulation modeling has shown that species can coevolve toward clusters of coexisting consumers exploiting the same limiting resource or resources, with nearly identical ratios of coefficients related to growth and mortality. This paper provides a mathematical basis for such as situation; a full analysis of the global dynamics of a new model for such a class of n-dimensional consumer–resource system, in which a set of consumers with identical growth to mortality ratios compete for the same resource and in which each consumer is mutualistic with the resource. First, we study the system of one resource and two consumers. By theoretical analysis, we demonstrate the expected result that competitive exclusion of one of the consumers can occur when the growth to mortality ratios differ. However, when these ratios are identical, the outcomes are complex. Either equilibrium coexistence or mutual extinction can occur, depending on initial conditions. When there is coexistence, interaction outcomes between the consumers can transition between effective mutualism, parasitism, competition, amensalism and neutralism. We generalize to the global dynamics of a system of one resource and multiple consumers. Changes in one factor, either a parameter or initial density, can determine whether all of the consumers either coexist or go to extinction together. New results are presented showing that multiple competing consumers can coexist on a single resource when they have coevolved toward identical growth to mortality ratios. This coexistence can occur because of feedbacks created by all of the consumers providing amutualistic service to the resource. This is biologically relevant to the persistence of pollination–mutualisms

Protocol selection for second-order consensus against disturbance

Noticing that both the absolute and relative velocity protocols can solve the second-order consensus of multi-agent systems, this paper aims to investigate which of the above two protocols has better anti-disturbance capability, in which the anti-disturbance capability is measured by the L2 gain from the disturbance to the consensus error. More specifically, by the orthogonal transformation technique, the analytic expression of the L2 gain of the second-order multi-agent system with absolute velocity protocol is firstly derived, followed by the counterpart with relative velocity protocol. It is shown that both the L2 gains for absolute and relative velocity protocols are determined only by the minimum non-zero eigenvalue of Laplacian matrix and the tunable gains of the state and velocity. Then, we establish the graph conditions to tell which protocol has better anti-disturbance capability. Moreover, we propose a two-step scheme to improve the anti-disturbance capability of second-order multi-agent systems. Finally, simulations are given to illustrate the effectiveness of our findings