Memory and mutualism in species sustainability: A time-fractional Lotka-Volterra model with harvesting

A fractional-order Wilson-Cowan formulation of cortical disinhibition

This work presents a fractional-order Wilson-Cowan model derivation under Caputo's formalism, considering an order of 0 < α ≤ 1 . To that end, we propose memory-dependent response functions and average neuronal excitation functions that permit us to naturally arrive at a fractional-order model that incorporates past dynamics into the description of synaptically coupled neuronal populations' activity. We then shift our focus on a particular example, aiming to analyze the fractional-order dynamics of the disinhibited cortex. This system mimics cortical activity observed during neurological disorders such as epileptic seizures, where an imbalance between excitation and inhibition is present, which allows brain dynamics to transition to a hyperexcited activity state. In the context of the first-order mathematical model, we recover traditional results showing a transition from a low-level activity state to a potentially pathological high-level activity state as an external factor modifies cortical inhibition. On the other hand, under the fractional-order formulation, we establish novel results showing that the system resists such transition as the order is decreased, permitting the possibility of staying in the low-activity state even with increased disinhibition. Furthermore, considering the memory index interpretation of the fractional-order model motivation here developed, our results establish that by increasing the memory index, the system becomes more resistant to transitioning towards the high-level activity state. That is, one possible effect of the memory index is to stabilize neuronal activity. Noticeably, this neuronal stabilizing effect is similar to homeostatic plasticity mechanisms. To summarize our results, we present a two-parameter structural portrait describing the system's dynamics dependent on a proposed disinhibition parameter and the order. We also explore numerical model simulations to validate our results.

Towards a sustainable fisher-dolphin coexistence: Understanding depredation, assessing economic damage and evaluating management options

Finding solutions for a sustainable coexistence between wildlife and humans is considered among the most challenging environmental management issues for scientists, conservationists, managers, and stockholders world-wide. Depredation by the common bottlenose dolphin (Tursiops truncatus) on small scale fisheries has increased in the recent years, leading to a growing conflict in many areas of the Mediterranean Sea and pressing for urgent management solutions. This study aims at developing a management framework for a sustainable coexistence between fishers and dolphins in Sardinia (Mediterranean Sea). Relying on the combination of different approaches (field study, literature review and Multi Criteria Decision Analysis), the scientific evidence necessary for understanding dolphin depredation were updated and improved, the related economic damage was calculated, and different management options were identified and evaluated by several stakeholder groups to support the decision-making process. Averaging for all investigated net types (gillnet and trammel net), a depredation frequency of 53% was found, the highest values ever found in both Sardinia and many other Mediterranean sites. Depredation probability was influenced by different factors, such as net type, fishing operation duration, depth of the fishing site and period. The estimated economic damage due to depredation ranges on average between 6492 and 11,925 euro per year and depends on the type of fishing net. The results from the field study, the literature review and the stakeholder involvement allowed us to define the most plausible and shared management options, identifying a framework for assessing and managing the conflict between fishers and dolphins for the creation of a more sustainable vision for the future.

Stability analysis of fractional order memristor synapse-coupled hopfield neural network with ring structure

A memristor is a nonlinear two-terminal electrical element that incorporates memory features and nanoscale properties, enabling us to design very high-density artificial neural networks. To enhance the memory property, we should use mathematical frameworks like fractional calculus, which is capable of doing so. Here, we first present a fractional-order memristor synapse-coupling Hopfield neural network on two neurons and then extend the model to a neural network with a ring structure that consists of n sub-network neurons, increasing the synchronization in the network. Necessary and sufficient conditions for the stability of equilibrium points are investigated, highlighting the dependency of the stability on the fractional-order value and the number of neurons. Numerical simulations and bifurcation analysis, along with Lyapunov exponents, are given in the two-neuron case that substantiates the theoretical findings, suggesting possible routes towards chaos when the fractional order of the system increases. In the n-neuron case also, it is revealed that the stability depends on the structure and number of sub-networks.

A Bi-Geometric Fractional Model for the Treatment of Cancer Using Radiotherapy

Our study is based on the modification of a well-known predator-prey equation, or the Lotka–Volterra competition model. That is, a system of differential equations was established for the population of healthy and cancerous cells within the tumor tissue of a patient struggling with cancer. Besides, fractional differentiation remedies the situation by obtaining a meticulous model with more flexible parameters. Furthermore, a specific type of non-Newtonian calculus, bi-geometric calculus, can describe the model in terms of proportions and implies the alternative aspect of a dynamic system. Moreover, fractional operators in bi-geometric calculus are formulated in terms of Hadamard fractional operators. In this article, the development of fractional operators in non-Newtonian calculus was investigated. The model was extended in these criteria, and the existence and uniqueness of the model were considered and guaranteed in the first step by applying the Arzelà–Ascoli. The bi-geometric analogue of the numerical method provided a suitable tool to solve the model approximately. In the end, the visual graphs were obtained by using the MATLAB program.

A Fractional-in-Time Prey–Predator Model with Hunting Cooperation: Qualitative Analysis, Stability and Numerical Approximations

A prey–predator system with logistic growth of prey and hunting cooperation of predators is studied. The introduction of fractional time derivatives and the related persistent memory strongly characterize the model behavior, as many dynamical systems in the applied sciences are well described by such fractional-order models. Mathematical analysis and numerical simulations are performed to highlight the characteristics of the proposed model. The existence, uniqueness and boundedness of solutions is proved; the stability of the coexistence equilibrium and the occurrence of Hopf bifurcation is investigated. Some numerical approximations of the solution are finally considered; the obtained trajectories confirm the theoretical findings. It is observed that the fractional-order derivative has a stabilizing effect and can be useful to control the coexistence between species.

Three-Species Lotka-Volterra Model with Respect to Caputo and Caputo-Fabrizio Fractional Operators

In this paper, we apply the concept of fractional calculus to study three-dimensional Lotka-Volterra differential equations. We incorporate the Caputo-Fabrizio fractional derivative into this model and investigate the existence of a solution. We discuss the uniqueness of the solution and determine under what conditions the model offers a unique solution. We prove the stability of the nonlinear model and analyse the properties, considering the non-singular kernel of the Caputo-Fabrizio operator. We compare the stability conditions of this system with respect to the Caputo-Fabrizio operator and the Caputo fractional derivative. In addition, we derive a new numerical method based on the Adams-Bashforth scheme. We show that the type of differential operators and the value of orders significantly influence the stability of the Lotka-Volterra system and numerical results demonstrate that different fractional operator derivatives of the nonlinear population model lead to different dynamical behaviors.

Fractional Lotka-Volterra-Type Cooperation Models: Impulsive Control on Their Stability Behavior

We present a biological fractional n-species delayed cooperation model of Lotka-Volterra type. The considered fractional derivatives are in the Caputo sense. Impulsive control strategies are applied for several stability properties of the states, namely Mittag-Leffler stability, practical stability and stability with respect to sets. The proposed results extend the existing stability results for integer-order n-species delayed Lotka-Volterra cooperation models to the fractional-order case under impulsive control.