Complex Dynamics in Predator-prey Models with Nonmonotonic Functional Response and Seasonal Harvesting

In this paper we study the complex dynamics of predator-prey systems with nonmonotonic functional response and harvesting. When the harvesting is constant-yield for prey, it is shown that various kinds of bifurcations, such as saddle-node bifurcation, degenerate Hopf bifurcation, and Bogdanov-Takens bifurcation, occur in the model as parameters vary. The existence of two limit cycles and a homoclinic loop is established by numerical simulations. When the harvesting is seasonal for both species, sufficient conditions for the existence of an asymptotically stable periodic solution and bifurcation of a stable periodic orbit into a stable invariant torus of the model are given. Numerical simulations are carried out to demonstrate the existence of bifurcation of a stable periodic orbit into an invariant torus and transition from invariant tori to periodic solutions, respectively, as the amplitude of seasonal harvesting increases

Multiple Attractors, Catastrophes and Chaos in Seasonally Perturbed Predator-Prey Communities

The classical predator-prey model is considered in this paper with reference to the case of periodically varying parameters. Six elementary seasonality mechanisms are identified and analyzed in detail by means of a continuation technique producing complete bifurcation diagrams. The results show that each elementary mechanism can give rise to multiple attractors and that catastrophic transitions can occur when suitable parameters are slightly changed. Moreover, the two classical routes to chaos, namely, torus destruction and cascade of period doublings, are numerically detected. Since in the case of constant parameters the model cannot have multiple attractors, catastrophes, and chaos, the results support the conjecture that seasons can very easily give rise to complex population dynamics

Seasons and Chaos in Ecosystems

This research report combines two journal articles on the relationships between seasons and chaos in ecosystems. They show that the strength of the seasons (i.e., the latitude) is a key factor for understanding the strange behavior of the ecosystem, and that chaos can be present in an assembly of different communities where the rhythm of the seasons suitably interferes with the endogenous rhythms of the biological processes. The first paper by Rinaldi et al. studies a general predator-prey model describing the behavior of two interacting populations in a periodic environment. Multiple attractors and catastrophic transitions are proved to exist and the two classical routes to chaos (torn destruction and cascade of period doublings) are numerically detected. The second paper by Doveri et al. presents a seasonally perturbed plankton-fish model composed by five compartments: nutrient, algae, zoo plankton, young fish, and adult fish. The bifurcation analysis of the model supports the conclusion that the dynamics of plankton communities can easily be chaotic provided that the strength of the season is sufficiently strong. In particular, the paper shows why large year-to-year differences in young fish survival need not always be attributable to external factors like interannual weather variability

Dynamics of Phytoplankton, Zooplankton and Fishery Resource Model

In this paper, a new mathematical model has been proposed and analyzed to study the interaction of phytoplankton- zooplankton-fish population in an aquatic environment with Holloing’s types II, III and IV functional responses. It is assumed that the growth rate of phytoplankton depends upon the constant level of nutrient and the fish population is harvested according to CPUE (catch per unit effort) hypothesis. Biological and bionomical equilibrium of the system has been investigated. Using Pontryagin’s Maximum Principal, the optimal harvesting policy is discussed. Chaotic nature and bifurcation analysis of the model system for a control parameter have been observed through a numerical simulation

Bifurcation analysis of Leslie-Gower predator-prey system with harvesting and fear effect

In the paper, a Leslie-Gower predator-prey system with harvesting and fear effect is considered. The existence and stability of all possible equilibrium points are analyzed. The bifurcation dynamic behavior at key equilibrium points is investigated to explore the intrinsic driving mechanisms of population interaction modes. It is shown that the system undergoes various bifurcations, including transcritical, saddle-node, Hopf and Bogdanov-Takens bifurcations. The numerical simulation results show that harvesting and fear effect can seriously affect the dynamic evolution trend and coexistence mode. Furthermore, it is particularly worth pointing out that harvesting not only drives changes in population coexistence mode, but also has a certain degree delay. Finally, it is anticipated that these research results will be beneficial for the vigorous development of predator-prey system

Connectivity, Complexity and Catastrophe in Large-scale Systems

This book represents an approach to large-scale system modeling that is a challenging synthesis for the systems analyst, the operations research worker, the system theorist, the policy analyst, and the student of social systems. After pointing out that the mathematical form of a system description dictates the types of questions that can be asked and answered by the model, the author declares that "there is no such thing as a model system: there are many models, each with its own characteristic mathematical features and each capable of addressing a certain subset of important questions about the system and its operation". The book supports this point with examples from a wide spectrum of contexts (such as physics, economic activity, water-resource management, ecology, transportation, and physiology) viewed from the points of view of various models and theories (such as general system theory, control theory, graph theory, linear and nonlinear system theory, and catastrophy theory). Against this broad background, the book then considers in depth the relations to large-scale systems of the theories of connectivity, complexity, stability, catastrophy, and resilience

Bifurcations of Invariant Tori in Predator-Prey Models with Seasonal Prey Harvesting

In this paper we study bifurcations in predator-prey systems with seasonal prey harvesting. First, when the seasonal harvesting reduces to constant yield, it is shown that various kinds of bifurcations, including saddle-node bifurcation, degenerate Hopf bifurcation, and Bogdanov--Takens bifurcation (i.e., cusp bifurcation of codimension 2), occur in the model as parameters vary. The existence of two limit cycles and a homoclinic loop is established. Bifurcation diagrams and phase portraits of the model are also given by numerical simulations, which reveal far richer dynamics compared to the case without harvesting. Second, when harvesting is seasonal (described by a periodic function), sufficient conditions for the existence of an asymptotically stable periodic solution and bifurcation of a stable periodic orbit into a stable invariant torus of the model are given. Numerical simulations, including bifurcation diagrams, phase portraits, and attractors of Poincaré maps, are carried out to demonstrate the existence of bifurcation of a stable periodic orbit into an invariant torus and bifurcation of a stable homoclinic loop into an invariant homoclinic torus, respectively, as the amplitude of seasonal harvesting increases. Our study indicates that to have persistence of the interacting species with seasonal harvesting in the form of asymptotically stable periodic solutions or stable quasi-periodic solutions, initial species densities should be located in the attraction basin of the hyperbolic stable equilibrium, stable limit cycle, or stable homoclinic loop, respectively, for the model with no harvesting or with constant-yield harvesting. Our study also demonstrates that the dynamical behaviors of the model are very sensitive to the constant-yield or seasonal prey harvesting, and careful management of resources and harvesting policies is required in the applied conservation and renewable resource contexts

Using MapReduce Streaming for Distributed Life Simulation on the Cloud

Distributed software simulations are indispensable in the study of large-scale life models but often require the use of technically complex lower-level distributed computing frameworks, such as MPI. We propose to overcome the complexity challenge by applying the emerging MapReduce (MR) model to distributed life simulations and by running such simulations on the cloud. Technically, we design optimized MR streaming algorithms for discrete and continuous versions of Conway’s life according to a general MR streaming pattern. We chose life because it is simple enough as a testbed for MR’s applicability to a-life simulations and general enough to make our results applicable to various lattice-based a-life models. We implement and empirically evaluate our algorithms’ performance on Amazon’s Elastic MR cloud. Our experiments demonstrate that a single MR optimization technique called strip partitioning can reduce the execution time of continuous life simulations by 64%. To the best of our knowledge, we are the first to propose and evaluate MR streaming algorithms for lattice-based simulations. Our algorithms can serve as prototypes in the development of novel MR simulation algorithms for large-scale lattice-based a-life models.https://digitalcommons.chapman.edu/scs_books/1014/thumbnail.jp