Numerical Continuation and Bifurcation Analysis in a Harvested Predator-Prey Model with Time Delay using DDE-Biftool

Time delay has been incorporated in models to reflect certain physical or biological meaning. The theory of delay differential equations (DDEs), which has seen extensive growth in the last seventy years or so, can be used to examine the effects of time delay in the dynamical behavior of systems being considered. Numerical tools to study DDEs have played a significant role not only in illustrating theoretical results but also in discovering interesting dynamics of the model. DDE-Biftool, which is a Matlab package for numerical continuation and numerical bifurcation analysis of DDEs, is one of the most utilized and popular numerical tools for DDEs. In this paper, we present a guide to using the latest version of DDE-Biftool targeted to researchers who are new to the study of time delay systems. A short discussion of an example application, which is a harvested predator-prey model with a single discrete time delay, will be presented first. We then implement this example model in DDE-Biftool, pointing out features where beginners need to be cautious. We end with a comparison of our theoretical and numerical results

Mechanisms explaining transitions between tonic and phasic firing in neuronal populations as predicted by a low dimensional firing rate model

Several firing patterns experimentally observed in neural populations have been successfully correlated to animal behavior. Population bursting, hereby regarded as a period of high firing rate followed by a period of quiescence, is typically observed in groups of neurons during behavior. Biophysical membrane-potential models of single cell bursting involve at least three equations. Extending such models to study the collective behavior of neural populations involves thousands of equations and can be very expensive computationally. For this reason, low dimensional population models that capture biophysical aspects of networks are needed. \noindent The present paper uses a firing-rate model to study mechanisms that trigger and stop transitions between tonic and phasic population firing. These mechanisms are captured through a two-dimensional system, which can potentially be extended to include interactions between different areas of the nervous system with a small number of equations. The typical behavior of midbrain dopaminergic neurons in the rodent is used as an example to illustrate and interpret our results. \noindent The model presented here can be used as a building block to study interactions between networks of neurons. This theoretical approach may help contextualize and understand the factors involved in regulating burst firing in populations and how it may modulate distinct aspects of behavior.Comment: 25 pages (including references and appendices); 12 figures uploaded as separate file

Continuation for thin film hydrodynamics and related scalar problems

This chapter illustrates how to apply continuation techniques in the analysis of a particular class of nonlinear kinetic equations that describe the time evolution through transport equations for a single scalar field like a densities or interface profiles of various types. We first systematically introduce these equations as gradient dynamics combining mass-conserving and nonmass-conserving fluxes followed by a discussion of nonvariational amendmends and a brief introduction to their analysis by numerical continuation. The approach is first applied to a number of common examples of variational equations, namely, Allen-Cahn- and Cahn-Hilliard-type equations including certain thin-film equations for partially wetting liquids on homogeneous and heterogeneous substrates as well as Swift-Hohenberg and Phase-Field-Crystal equations. Second we consider nonvariational examples as the Kuramoto-Sivashinsky equation, convective Allen-Cahn and Cahn-Hilliard equations and thin-film equations describing stationary sliding drops and a transversal front instability in a dip-coating. Through the different examples we illustrate how to employ the numerical tools provided by the packages auto07p and pde2path to determine steady, stationary and time-periodic solutions in one and two dimensions and the resulting bifurcation diagrams. The incorporation of boundary conditions and integral side conditions is also discussed as well as problem-specific implementation issues

Inverse bifurcation analysis: application to simple gene systems

BACKGROUND: Bifurcation analysis has proven to be a powerful method for understanding the qualitative behavior of gene regulatory networks. In addition to the more traditional forward problem of determining the mapping from parameter space to the space of model behavior, the inverse problem of determining model parameters to result in certain desired properties of the bifurcation diagram provides an attractive methodology for addressing important biological problems. These include understanding how the robustness of qualitative behavior arises from system design as well as providing a way to engineer biological networks with qualitative properties. RESULTS: We demonstrate that certain inverse bifurcation problems of biological interest may be cast as optimization problems involving minimal distances of reference parameter sets to bifurcation manifolds. This formulation allows for an iterative solution procedure based on performing a sequence of eigen-system computations and one-parameter continuations of solutions, the latter being a standard capability in existing numerical bifurcation software. As applications of the proposed method, we show that the problem of maximizing regions of a given qualitative behavior as well as the reverse engineering of bistable gene switches can be modelled and efficiently solved

Controlling multistability in a vibro-impact capsule system

This is the final version of the article. Available from Springer Verlag via the DOI in this record.This work concerns the control of multistability in a vibro-impact capsule system driven by a harmonic excitation. The capsule is able to move forward and backward in a rectilinear direction, and the main objective of this work is to control such motion in the presence of multiple coexisting periodic solutions. A position feedback controller is employed in this study, and our numerical investigation demonstrates that the proposed control method gives rise to a dynamical scenario with two coexisting solutions, corresponding to forward and backward progression. Therefore, the motion direction of the system can be controlled by suitably perturbing its initial conditions, without altering the system parameters. To study the robustness of this control method, we apply numerical continuation methods in order to identify a region in the parameter space in which the proposed controller can be applied. For this purpose, we employ the MATLAB-based numerical platform COCO, which supports the continuation and bifurcation detection of periodic orbits of non-smooth dynamical systems.The second author has been supported by a Georg Forster Research Fellowship granted by the Alexander von Humboldt Foundation, Germany. The authors would like to thank Dr. Haibo Jiang for stimulating discussions and comments on this work

Modelling Cell Polarization Driven by Synthetic Spatially Graded Rac Activation

The small GTPase Rac is known to be an important regulator of cell polarization, cytoskeletal reorganization, and motility of mammalian cells. In recent microfluidic experiments, HeLa cells endowed with appropriate constructs were subjected to gradients of the small molecule rapamycin leading to synthetic membrane recruitment of a Rac activator and direct graded activation of membrane-associated Rac. Rac activation could thus be triggered independent of upstream signaling mechanisms otherwise responsible for transducing activating gradient signals. The response of the cells to such stimulation depended on exceeding a threshold of activated Rac. Here we develop a minimal reaction-diffusion model for the GTPase network alone and for GTPase-phosphoinositide crosstalk that is consistent with experimental observations for the polarization of the cells. The modeling suggests that mutual inhibition is a more likely mode of cell polarization than positive feedback of Rac onto its own activation. We use a new analytical tool, Local Perturbation Analysis, to approximate the partial differential equations by ordinary differential equations for local and global variables. This method helps to analyze the parameter space and behaviour of the proposed models. The models and experiments suggest that (1) spatially uniform stimulation serves to sensitize a cell to applied gradients. (2) Feedback between phosphoinositides and Rho GTPases sensitizes a cell. (3) Cell lengthening/flattening accompanying polarization can increase the sensitivity of a cell and stabilize an otherwise unstable polarization

Mitochondrial chaotic dynamics: Redox-energetic behavior at the edge of stability

Mitochondria serve multiple key cellular functions, including energy generation, redox balance, and regulation of apoptotic cell death, thus making a major impact on healthy and diseased states. Increasingly recognized is that biological network stability/instability can play critical roles in determining health and disease. We report for the first-time mitochondrial chaotic dynamics, characterizing the conditions leading from stability to chaos in this organelle. Using an experimentally validated computational model of mitochondrial function, we show that complex oscillatory dynamics in key metabolic variables, arising at the “edge” between fully functional and pathological behavior, sets the stage for chaos. Under these conditions, a mild, regular sinusoidal redox forcing perturbation triggers chaotic dynamics with main signature traits such as sensitivity to initial conditions, positive Lyapunov exponents, and strange attractors. At the “edge” mitochondrial chaos is exquisitely sensitive to the antioxidant capacity of matrix Mn superoxide dismutase as well as to the amplitude and frequency of the redox perturbation. These results have potential implications both for mitochondrial signaling determining health maintenance, and pathological transformation, including abnormal cardiac rhythms.publishedVersionKembro, Jackelyn Melissa. Universidad Nacional de Córdoba. Facultad de Ciencias Exactas, Físicas y Naturales; Argentina.Kembro, Jackelyn Melissa. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto de Investigaciones Biológicas y Tecnológicas; Argentina.Cortassa, Sonia. National Institutes of Health. NIH · NIA Intramural Research Program; Estados Unidos.Lloyd, David. Cardiff University. School of Biosciences 1; Inglaterra.Sollot, Steven. Johns Hopkins University. Laboratory of Cardiovascular Science; Estados Unidos.Sollot, Steven. Johns Hopkins University. Laboratory of Cardiovascular Science; Estados Unidos

Does Sex-Selective Predation Stabilize or Destabilize Predator-Prey Dynamics?

Background: Little is known about the impact of prey sexual dimorphism on predator-prey dynamics and the impact of sexselective harvesting and trophy hunting on long-term stability of exploited populations. Methodology and Principal Findings: We review the quantitative evidence for sex-selective predation and study its longterm consequences using several simple predator-prey models. These models can be also interpreted in terms of feedback between harvesting effort and population size of the harvested species under open-access exploitation. Among the 81 predator-prey pairs found in the literature, male bias in predation is 2.3 times as common as female bias. We show that long-term effects of sex-selective predation depend on the interplay of predation bias and prey mating system. Predation on the ‘less limiting’ prey sex can yield a stable predator-prey equilibrium, while predation on the other sex usually destabilizes the dynamics and promotes population collapses. For prey mating systems that we consider, males are less limiting except for polyandry and polyandrogyny, and male-biased predation alone on such prey can stabilize otherwise unstable dynamics. On the contrary, our results suggest that female-biased predation on polygynous, polygynandrous or monogamous prey requires other stabilizing mechanisms to persist. Conclusions and Significance: Our modelling results suggest that the observed skew towards male-biased predation might reflect, in addition to sexual selection, the evolutionary history of predator-prey interactions. More focus on these phenomena can yield additional and interesting insights as to which mechanisms maintain the persistence of predator-prey pairs over ecological and evolutionary timescales. Our results can also have implications for long-term sustainability of harvesting and trophy hunting of sexually dimorphic species