On a numerical bifurcation analysis of a particle reaction-diffusion model for a motion of two self-propelled disks

Theoretical analysis using mathematical models is often used to understand a mechanism of collective motion in a self-propelled system. In the experimental system using camphor disks, several kinds of characteristic motions have been observed due to the interaction of two camphor disks. In this paper, we understand the emergence mechanism of the motions caused by the interaction of two self-propelled bodies by analyzing the global bifurcation structure using the numerical bifurcation method for a mathematical model. Finally, it is also shown that the irregular motion, which is one of the characteristic motions, is chaotic motion and that it arises from periodic bifurcation phenomena and quasi-periodic motions due to torus bifurcation.

Dynamic analysis of a malaria reaction-diffusion model with periodic delays and vector bias

<abstract><p>One of the most important vector-borne disease in humans is malaria, caused by <italic>Plasmodium</italic> parasite. Seasonal temperature elements have a major effect on the life development of mosquitoes and the development of parasites. In this paper, we establish and analyze a reaction-diffusion model, which includes seasonality, vector-bias, temperature-dependent extrinsic incubation period (EIP) and maturation delay in mosquitoes. In order to get the model threshold dynamics, a threshold parameter, the basic reproduction number $ R_{0} $ is introduced, which is the spectral radius of the next generation operator. Quantitative analysis indicates that when $ R_{0} &lt; 1 $, there is a globally attractive disease-free $ \omega $-periodic solution; disease is uniformly persistent in humans and mosquitoes if $ R_{0} &gt; 1 $. Numerical simulations verify the results of the theoretical analysis and discuss the effects of diffusion and seasonality. We study the relationship between the parameters in the model and $ R_{0} $. More importantly, how to allocate medical resources to reduce the spread of disease is explored through numerical simulations. Last but not least, we discover that when studying malaria transmission, ignoring vector-bias or assuming that the maturity period is not affected by temperature, the risk of disease transmission will be underestimate.</p></abstract>

Impulsive Reaction-Diffusion Delayed Models in Biology: Integral Manifolds Approach

In this paper we study an impulsive delayed reaction-diffusion model applied in biology. The introduced model generalizes existing reaction-diffusion delayed epidemic models to the impulsive case. The integral manifolds notion has been introduced to the model under consideration. This notion extends the single state notion and has important applications in the study of multi-stable systems. By means of an extension of the Lyapunov method integral manifolds’ existence, results are established. Based on the Lyapunov functions technique combined with a Poincarè-type inequality qualitative criteria related to boundedness, permanence, and stability of the integral manifolds are also presented. The application of the proposed impulsive control model is closely related to a most important problems in the mathematical biology—the problem of optimal control of epidemic models. The considered impulsive effects can be used by epidemiologists as a very effective therapy control strategy. In addition, since the integral manifolds approach is relevant in various contexts, our results can be applied in the qualitative investigations of many problems in the epidemiology of diverse interest.

Markov Chain Abstractions of Electrochemical Reaction-Diffusion in Synaptic Transmission for Neuromorphic Computing

Progress in computational neuroscience toward understanding brain function is challenged both by the complexity of molecular-scale electrochemical interactions at the level of individual neurons and synapses and the dimensionality of network dynamics across the brain covering a vast range of spatial and temporal scales. Our work abstracts an existing highly detailed, biophysically realistic 3D reaction-diffusion model of a chemical synapse to a compact internal state space representation that maps onto parallel neuromorphic hardware for efficient emulation at a very large scale and offers near-equivalence in input-output dynamics while preserving biologically interpretable tunable parameters.

Synchronization of the Glycolysis Reaction-Diffusion Model via Linear Control Law

The Selkov system, which is typically employed to model glycolysis phenomena, unveils some rich dynamics and some other complex formations in biochemical reactions. In the present work, the synchronization problem of the glycolysis reaction-diffusion model is handled and examined. In addition, a novel convenient control law is designed in a linear form and, on the other hand, the stability of the associated error system is demonstrated through utilizing a suitable Lyapunov function. To illustrate the applicability of the proposed schemes, several numerical simulations are performed in one- and two-spatial dimensions.

Effect of delay on pattern formation of a Rosenzweig–MacArthur type reaction–diffusion model with spatiotemporal delay

In this paper, a predator–prey reaction–diffusion model with Rosenzweig–MacArthur type functional response and spatiotemporal delay is investigated through using the tool of Turing bifurcation theories. First, by taking the average time delay as a bifurcation parameter, conditions of occurrence of Turing bifurcation are obtained through employing the Routh–Hurwitz criteria. Second, as the average time delay varies the amplitude equations of Turing bifurcation patterns including spots pattern and stripes pattern are also obtained through the multiple scale perturbation method. Finally, the two kinds of spatiotemporal evolution distributions of species such as spots pattern and stripes pattern are shown to illustrate theoretical results.

A reaction-diffusion model predicts the intracellular length scale over which EGFR-initiated GAB1-SHP2 complexes persist

Activation of receptor tyrosine kinases (RTKs) leads to the assembly of multi-membered protein complexes connected by phosphotyrosine-SH2 domain linkages. However, these linkages are relatively weak and reversible, which allows complex disassembly to occur on a time scale that permits phosphatases to dephosphorylate complex members and thereby regulate complex persistence. Here, we generated a computational reaction-diffusion model to predict the length scale over which membrane-bound RTKs can regulate the maintenance of such protein complexes through the intermediary action of diffusible cytoplasmic kinases. Specifically, we show that the RTK EGFR can activate SRC family kinases (SFKs) to maintain the association of SHP2 with phosphorylated GAB1, which activates SHP2, throughout the entire cell volume. This finding is dependent on the ability of SFKs to be activated by EGFR at the plasma membrane and subsequently diffuse through the cytosol, as altering the model topology to permit only SFK activation at the plasma membrane reduces the length scale of GAB1-SHP2 association. Modifying the model topology to neglect GAB1 binding to cytosolic and EGFR-bound GRB2 had little effect on this length scale. Indeed, a model sensitivity analysis identified protein diffusion, SFK inactivation, and GAB1 dephosphorylation as the processes that most strongly control the distance over which GAB1-SHP2 persists distal from EGFR. A model scaling analysis likewise predicted that the length scale of GAB1-SHP2 association is greatly extended compared to that of SFK activation and that GAB1-SHP2 complexes persist throughout the cell volume. Furthermore, the same processes identified in the model sensitivity analysis appeared in the length scale estimate for GAB1-SHP2 association. In vitro experiments using proximity ligation assay and immunofluorescence against GAB1-SHP2 and EGFR, respectively, suggested that GAB1-SHP2 complexes are distributed throughout cells and exist distally from EGFR during EGF stimulation. Overall, our results suggest that GAB1-SHP2 complexes—and thus active SHP2—can persist distally from EGFR due to re-phosphorylation of GAB1 throughout the cytosol by EGFR-activated SFKs.