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In this paper we study the permanence and impermanence for continuous-time competitive Kolmogorov systems via the carrying simplex. We first give an extension to attractors of V. Hutson's results on the existence of repellors in continuous-time dynamical systems that have found wide use in the study of permanence via average Liapunov functions. We then give a general criterion for the stability of the boundary of carrying simplex for competitive Kolmogorov systems of differential equations, which determines the permanence and impermanence of such systems. Based on the criterion, we present a complete classification of the permanence and impermanence in terms of inequalities on parameters for all three-dimensional competitive systems which have linearly determined nullclines. The results are applied to many classical models in population dynamics including the Lotka-Volterra system, Gompertz system, Leslie-Gower system and Ricker system. [ABSTRACT FROM AUTHOR]

In a previous paper (Feng et al., J. Differential Equations (2023)), we studied a seasonally interactive model between closed seasons and open seasons with Michaelis-Menten type harvesting, in which we assumed that the harvesting quantity was relatively large (0 < κ = l G c E < 1) and obtained a length threshold of the closed season T ¯ ⁎ depending on the harvesting parameter and the seasonal fluctuation period. It was shown that the origin is globally asymptotically stable if and only if T ¯ ≤ T ¯ ⁎ , and there exists a unique globally asymptotically stable T -periodic solution if and only if T ¯ > T ¯ ⁎. In this paper, we continue to investigate the periodic dynamics of this model when the harvesting quantity is relatively small (κ = l G c E ≥ 1). By finding another smaller length threshold T ¯ ⁎ ∈ (0 , T ¯ ⁎) , we determine the number of periodic solutions and study their stability, which imply the occurrence of bifurcation of periodic solutions. Our results demonstrate that designing the closed harvesting season properly can prevent the species from extinction. Moreover, by comparing with the continuous harvesting model and combining the results in our previous paper and this article, we provide a complete understanding on the global dynamics for the periodic switching model. Some numerical simulations are also carried out to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract Volterra integro-differential equations with kernels of integral operators representable by Stieltjes integrals are investigated. The presented results are based on the approach related to the study of one-parameter semigroups for linear evolution equations. We present the method of reduction of the original initial-value problem for a model integro-differential equation with operator coefficients in a Hilbert space to the Cauchy problem for a first-order differential equation in an extended function space. The existence of a contractive C0-semigroup is proved. An estimate for the exponential decay of the semigroup is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we consider n-type Markov branching–immigration processes. The uniqueness criterion is first established. Then, we construct a related system of differential equations based on the branching property. Furthermore, the explicit expression of extinction probability and the mean extinction time are successfully obtained in the absorbing case by using the unique solution of the related system of differential equations and Kolmogorov forward equations. Finally, the recurrence and ergodicity criteria are given if the zero state 0 is not absorbing. [ABSTRACT FROM AUTHOR]

Sufficient conditions are established for the boundedness of all solutions and their first two derivatives of a third-order linear integro-differential equation of the Volterra type on the half-line. To this end, using a method proposed by the first author in 2006, first, we reduce the equation under consideration to an equivalent system consisting of one first-order differential equation and one second-order Volterra integro-differential equation. Then a new generalized Lyapunov functional is proposed for this system, the nonnegativity of this functional on solutions of this system is proved, and an upper bound is given for the derivative of this functional via the original functional. The resulting estimate is an integro-differential inequality whose solution gives an estimate of the functional. [ABSTRACT FROM AUTHOR]

Mathematical modelling of biological population dynamics often involves proposing high-fidelity discrete agent-based models that capture stochasticity and individual-level processes. These models are often considered in conjunction with an approximate coarse-grained differential equation that captures population-level features only. These coarse-grained models are only accurate in certain asymptotic parameter regimes, such as enforcing that the time scale of individual motility far exceeds the time scale of birth/death processes. When these coarse-grained models are accurate, the discrete model still abides by conservation laws at the microscopic level, which implies that there is some macroscopic conservation law that can describe the macroscopic dynamics. In this work, we introduce an equation learning framework to find accurate coarse-grained models when standard continuum limit approaches are inaccurate. We demonstrate our approach using a discrete mechanical model of epithelial tissues, considering a series of four case studies that consider problems with and without free boundaries, and with and without proliferation, illustrating how we can learn macroscopic equations describing mechanical relaxation, cell proliferation, and the equation governing the dynamics of the free boundary of the tissue. While our presentation focuses on this biological application, our approach is more broadly applicable across a range of scenarios where discrete models are approximated by approximate continuum-limit descriptions. [ABSTRACT FROM AUTHOR]

In this article, the ratio-dependent prey-predator system perturbed with time noise is numerically investigated. It relates to the population densities of the prey and predator in an ecological system. The initial prey-predator models only depend on the time and a couple of the differential equations. We are considering a model where the prey-predator interaction is influenced by both space and time and the need for a coupled nonlinear partial differential equation with the effect of the random behavior of the environment. The existence of the solutions is guaranteed by using Schauder's fixed point theorem. The computation of the underlying model is carried out by two schemes. The proposed stochastic forward Euler scheme is conditionally stable and consistent with the system of the equations. The proposed stochastic non-standard finite difference scheme is unconditionally stable and consistent with the system of the equations. The graphical behavior of a test problem for different values of the parameters is shown which depicts the efficacy of the schemes. Our numerical results will help the researchers to consider the effect of the noise on the prey-predator model. [ABSTRACT FROM AUTHOR]

The aim of this work is to modify the invariant subspace method (ISM) in order to obtain closed form solutions of fractional differential-difference equations with Ψ-Riemann–Liouville (Ψ-RL) fractional derivative for first time. We have investigated the cases of two-dimensional and the three-dimensional invariant subspaces (ISs) in the suggested scheme. Using the modified ISM, new exact generalized solutions for the general fractional mKdV Lattice equation and the fractional Volterra lattice system are obtained. Compared with similar solution techniques in literature, the presented solution scheme is highly efficient and is capable to find new general exact solutions which cannot be attained by other methods. [ABSTRACT FROM AUTHOR]

Long COVID, a long-lasting disorder following an acute infection of COVID-19, represents a significant public health burden at present. In this paper, we propose a new mechanistic model based on differential equations to investigate the population dynamics of long COVID. By connecting long COVID with acute infection at the population level, our modeling framework emphasizes the interplay between COVID-19 transmission, vaccination, and long COVID dynamics. We conducted a detailed mathematical analysis of the model. We also validated the model using numerical simulation with real data from the US state of Tennessee and the UK. [ABSTRACT FROM AUTHOR]

In the realm of ecology, species naturally strive to enhance their own survival odds. This study introduces and investigates a predator-prey model incorporating reaction-diffusion through a system of differential equations. We scrutinize how diffusion impacts the model's stability. By analysing the stability of the model's uniform equilibrium state, we identify a condition leading to Turing instability. The study delves into how diffusion influences pattern formation within a predator-prey system. Our findings reveal that various spatiotemporal patterns, such as patches, spots, and even chaos, emerge based on species diffusion rates. We derive the amplitude equation by employing the weak nonlinear multiple scales analysis technique and the Taylor series expansion. A novel sinc interpolation approach is introduced. Numerical simulations elucidate the interplay between diffusion and Turing parameters. In a two-dimensional domain, spatial pattern analysis illustrates population density dynamics resulting in isolated groups, spots, stripes, or labyrinthine patterns. Simulation results underscore the method's effectiveness. The article concludes by discussing the biological implications of these outcomes. [ABSTRACT FROM AUTHOR]

In this paper, orthogonal fuzzy versions are reported for some celebrated iterative mappings. We provide various concrete conditions on the real valued functions J , S : (0 , 1 ] → (− ∞ , ∞) for the existence of fixed-points of (J , S) -fuzzy interpolative contractions. This way, many fixed point theorems are developed in orthogonal fuzzy metric spaces. We apply the (J , S) -fuzzy version of Banach fixed point theorem to demonstrate the existence and uniqueness of the solution. These results are supported with several non-trivial examples and applications to Volterra-type integral equations and fractional differential equations. [ABSTRACT FROM AUTHOR]

In this study, a conformable fractional order Lotka–Volterra predator-prey model that describes the COVID-19 dynamics is considered. By using a piecewise constant approximation, a discretization method, which transforms the conformable fractional-order differential equation into a difference equation, is introduced. Algebraic conditions for ensuring the stability of the equilibrium points of the discrete system are determined by using Schur–Cohn criterion. Bifurcation analysis shows that the discrete system exhibits Neimark–Sacker bifurcation around the positive equilibrium point with respect to changing the parameter d and e. Maximum Lyapunov exponents show the complex dynamics of the discrete model. In addition, the COVID-19 mathematical model consisting of healthy and infected populations is also studied on the Erdős Rényi network. If the coupling strength reaches the critical value, then transition from nonchaotic to chaotic state is observed in complex dynamical networks. Finally, it has been observed that the dynamical network tends to exhibit chaotic behavior earlier when the number of nodes and edges increases. All these theoretical results are interpreted biologically and supported by numerical simulations. [ABSTRACT FROM AUTHOR]

Physical laws governing population dynamics are generally expressed as differential equations. Research in recent decades has incorporated fractional‐order (non‐integer) derivatives into differential models of natural phenomena, such as reaction–diffusion systems. In this paper, we develop a method to numerically solve a multi‐component and multi‐dimensional space‐fractional system. For space discretization, we apply a Fourier spectral method that is suited for multidimensional partial differential equation systems. Efficient approximation of time‐stepping is accomplished with a locally one dimensional exponential time differencing approach. We show the effect of different fractional parameters on growth models and consider the convergence, stability, and uniqueness of solutions, as well as the biological interpretation of parameters and boundary conditions. [ABSTRACT FROM AUTHOR]

In this article, the ratio-dependent prey-predator system perturbed with time noise is numerically investigated. It relates to the population densities of the prey and predator in an ecological system. The initial prey-predator models only depend on the time and a couple of the differential equations. We are considering a model where the prey-predator interaction is influenced by both space and time and the need for a coupled nonlinear partial differential equation with the effect of the random behavior of the environment. The existence of the solutions is guaranteed by using Schauder's fixed point theorem. The computation of the underlying model is carried out by two schemes. The proposed stochastic forward Euler scheme is conditionally stable and consistent with the system of the equations. The proposed stochastic non-standard finite difference scheme is unconditionally stable and consistent with the system of the equations. The graphical behavior of a test problem for different values of the parameters is shown which depicts the efficacy of the schemes. Our numerical results will help the researchers to consider the effect of the noise on the prey-predator model. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of singular prey‐taxis models in a smooth bounded domain under homogeneous Neumann boundary conditions. The main challenge of analysis is the possible singularity as the prey density vanishes. Employing the technique of a priori assumption, the comparison principle of differential equations and semigroup estimates, we show that the singularity can be precluded if the intrinsic growth rate of prey is suitably large and hence obtain the existence of global classical bounded solutions. Moreover, the global stability of co‐existence and prey‐only steady states with convergence rates is established by the method of Lyapunov functionals. [ABSTRACT FROM AUTHOR]

The well known results of the theory of classical gaseous polytropes are presented in the framework of an integral approach where the standard Lane-Emden differential equation for a spherically symmetric gravitating mass is examined via its equivalent in the form of a nonlinear integral Volterra equation of the 2nd kind. It is shown that the inverse Laplace transform of the Lane-Emden equation for polytropes with an index of n=5 (Schuster model) is a recurrence relation for Bessel functions of the first kind. The invariance of the nonlinear integral Volterra equation with respect to homological transformations is shown, as well as the possibility of obtaining singular solutions under certain conditions. It is also shown that for whole integral and half integral polytrope indices, this equation is equivalent to a multidimensional integral equation, and finding the expansion of the Emden function in a power series of the dimensionless distance ξ from the center of the polytrope is equivalent to finding the Neumann series and the iterated nuclei in the Fredholm theory. Approximations of the Emden functions in closed form and their applicability to various astrophysical objects will be presented and discussed in the second part of this paper. Polytropes of other geometries and dimensionalities are not considered here. [ABSTRACT FROM AUTHOR]

In this paper, we analyze Levitan and Bebutov metrical approximations of functions F : Λ x X → Y by trigonometric polynomials and ρ-periodic type functions, where ∅= Λ ⊆ Rn, X and Y are complex Banach spaces, and ρ is a general binary relation on Y . We also analyze various classes of multidimensional Levitan almost periodic functions in general metric and multidimensional Bebutov uniformly recurrent functions in general metric. We provide several applications of our theoretical results to the abstract Volterra integro-differential equations and the partial differential equations. [ABSTRACT FROM AUTHOR]

This paper focuses on computational technique to solve linear systems of Volterra integro-fractional differential equations (LSVIFDEs) in the Caputo sense for all fractional order lins in 0 , 1 using two and three order block-by-block approach with explicit finite difference approximation. With this method, we aim to use an appropriate process to transform our problem into an analogous piecewise iterative linear algebraic system. Moreover, algorithms for treating LSVIFDEs using the above process have been developed, in order to express these solutions. In addition, numerical examples for our scheme are presented based on various kernels, including symmetry kernel and other sorts of separate kernels, are used to illustrate the validity, effectiveness and applicability of the suggested approach. Consequently, comparisons are performed with exact results using this technique, to communicate these approaches most general programs are written in Python V 3.8.8 software 2021 . [ABSTRACT FROM AUTHOR]

Allometric settings of population dynamics models are appealing due to their parsimonious nature and broad utility when studying system level effects. Here, we parameterise the size-scaled Rosenzweig-MacArthur differential equations to eliminate prey-mass dependency, facilitating an in depth analytic study of the equations which incorporates scaling parameters' contributions to coexistence. We define the functional response term to match empirical findings, and examine situations where metabolic theory derivations and observation diverge. The dynamical properties of the Rosenzweig-MacArthur system, encompassing the distribution of size-abundance equilibria, the scaling of period and amplitude of population cycling, and relationships between predator and prey abundances, are consistent with empirical observation. Our parameterisation is an accurate minimal model across 15+ orders of mass magnitude. [ABSTRACT FROM AUTHOR]

In this article, the ratio-dependent prey-predator system perturbed with time noise is numerically investigated. It relates to the population densities of the prey and predator in an ecological system. The initial prey-predator models only depend on the time and a couple of the differential equations. We are considering a model where the prey-predator interaction is influenced by both space and time and the need for a coupled nonlinear partial differential equation with the effect of the random behavior of the environment. The existence of the solutions is guaranteed by using Schauder's fixed point theorem. The computation of the underlying model is carried out by two schemes. The proposed stochastic forward Euler scheme is conditionally stable and consistent with the system of the equations. The proposed stochastic non-standard finite difference scheme is unconditionally stable and consistent with the system of the equations. The graphical behavior of a test problem for different values of the parameters is shown which depicts the efficacy of the schemes. Our numerical results will help the researchers to consider the effect of the noise on the prey-predator model. [ABSTRACT FROM AUTHOR]

This paper is in concern with the study of differential problems involving the ψ$$ \psi $$‐shifted fractional derivatives, where ψ$$ \psi $$ is a scaling function. Such operators can be thought of as a generalization of several fractional derivatives such as the classical Riemann–Liouville and Caputo operators, the Hadamard operators, the generalized fractional operators, and the Erdélyi–Kober operators, among others. Two types of boundary conditions are considered: the Cauchy problems and the integral boundary conditions. [ABSTRACT FROM AUTHOR]

The spectral properties of a linear operator that is the generator of a semigroup generated by a Volterra integro-differential equation in a Hilbert space are studied. Such integro-differential equations can be implemented as partial integro-differential equations arising in the theory of viscoelasticity and the theory of heat propagation in media with memory and also have many other important applications.The established results on the Riesz basis property of the root vectors of the semigroup generator can be used in studying the properties of solutions of integro-differential equations. [ABSTRACT FROM AUTHOR]

Predator-prey models are often used in ecology to represent realistic encounters between species. Lotka-Volterra differential equations are famously used by ecologists to model the relationship between a predator and prey. Given ecological data collection is known to be time consuming and difficult, the purpose of this paper is to optimize the process and improve sampling efficiency. The method of sequential optimality is applied to a more complex population model with multiple basins of attraction, the Lotka-Volterra differential equations. In this application, various theoretical scenarios are simulated using varying sample sizes and timeframes to determine optimal designs. The first scenario uses a standard sample size and design window to represent a realistic budget for ecological data collection. The second scenario decreases the budget and extends the design window as a representation of a limited budget. The third scenario represents ample resources available for sampling. The three scenarios are used to then compare the optimal designs and help ecologists evaluate the method of sequential optimality. I, A and D optimal designs are compared in the simulation results. The I-optimality criterion tends to outperform the A and D criteria and is recommended as a primary selection when using the sequential optimality algorithm. This simulation study has further developed sequential optimality by predicting optimal sampling regimes according to the dynamics associated with the Lotka-Volterra differential equations. [ABSTRACT FROM AUTHOR]

We extend the recently developed rough path theory for Volterra equations from [F. Harang and S. Tindel, Volterra equations driven by rough signals, Stoch. Process. Appl. 142 (2021) 34–78] to the case of more rough noise and/or more singular Volterra kernels. It was already observed in [F. Harang and S. Tindel, Volterra equations driven by rough signals, Stoch. Process. Appl. 142 (2021) 34–78] that the Volterra rough path introduced there did not satisfy any geometric relation, similar to that observed in classical rough path theory. Thus, an extension of the theory to more irregular driving signals requires a deeper understanding of the specific algebraic structure arising in the Volterra rough path. Inspired by the elements of "non-geometric rough paths" developed in [M. Gubinelli, Ramification of rough paths, J. Differential Equations 248 (2010) 693–721; M. Hairer and D. Kelly, Geometric versus non-geometric rough path, Ann. Inst. Henri Poincaré-Probab. Stat. 51 (2015) 207–251], we provide a simple description of the Volterra rough path and the controlled Volterra process in terms of rooted trees, and with this description we are able to solve rough Volterra equations driven by more irregular signals. [ABSTRACT FROM AUTHOR]

We consider a class of semi-linear differential Volterra equations with memory terms, polynomial nonlinearities and random perturbation. For a broad class of nonlinearities, we show that the system in concern admits a unique weak solution. Also, any statistically steady state must possess regularity compatible with that of the solution. Moreover, if sufficiently many directions are stochastically forced, we employ the generalized coupling approach to prove that there exists a unique invariant probability measure and that the system is exponentially attractive. This extends ergodicity results previously established in Bonaccorsi et al., (2012). [ABSTRACT FROM AUTHOR]

Consider a Lotka–Volterra (LV) system of coupled differential equations: x ̇ k = x k (r k − x k + (B x) k) , x = (x k) , 1 ≤ k ≤ n , where r = (r k) is a n × 1 vector and B a n × n matrix. Assume that the interaction matrix B is random and follows the elliptic model: B = 1 α n A + μ n 1 n 1 n T , where A = (A i j) is a n × n matrix with N (0 , 1) entries satisfying the following dependence structure (i) the entries A i j on and above the diagonal are i.i.d., (i i) for i < j each vector (A i j , A j i) is standard Gaussian with covariance ρ , and independent from the other entries; vector 1 n stands for the n × 1 vector of ones. Parameters α , μ are deterministic and may depend on n. Leveraging on Random Matrix Theory, we analyze this LV system as n → ∞ and study the existence of a positive equilibrium. This question boils down to study the existence of a (componentwise) positive solution to the linear equation: x = r + B x , depending on B 's parameters (α , μ , ρ) , a problem of independent interest in linear algebra. In the case where no positive equilibrium exists, we provide sufficient conditions for the existence of a unique stable equilibrium (with vanishing components), and following Bunin (2017), present a heuristics estimating the number of positive components of the equilibrium and their distribution. The existence of positive equilibria for large Lotka–Volterra systems has been raised in Dougoud et al. (2018), and addressed in various contexts by Bizeul and Najim (2021) and Akjouj and Najim (2022). Such LV systems are widely used in mathematical biology to model populations with interactions, and the existence of a positive equilibrium known as a feasible equilibrium corresponds to the survival of all the species x k within the system. [ABSTRACT FROM AUTHOR]

We present a new mixed finite element method for a class of parabolic equations with p -Laplacian and nonlinear memory. The applicability, stability and convergence of the method are studied. First, the problem is written in a mixed formulation as a system of one parabolic equation and a Volterra equation. Then, the system is discretized in the space variable using the finite element method with Lagrangian basis of degree r ≥ 1. Finally, the Cranck-Nicolson method with the trapezoidal quadrature is applied to discretize the time variable. For each method, we establish existence, uniqueness and regularity of the solutions. The convergence order is found to be dependent on the parameter p on the p -Laplacian in the sense that it decreases as p increases. [ABSTRACT FROM AUTHOR]

A numerical method based on the inverse Sumudu transform and the Bernstein polynomials operational matrix of integration is developed. The derived method is implemented in solving linear differential, integral and integro-differential equations. Also, a procedure for overcoming nonlinearity is developed and implemented to solve nonlinear Volterra integral equations. The approximate results are compared with the exact solutions and an existing method. Error estimation shows that the proposed method has elevated level of accuracy for just a few terms of the polynomial. [ABSTRACT FROM AUTHOR]

In this paper, we consider an autonomous two-dimensional ODE Kolmogorov-type system with three parameters, which is a particular system of the general predator–prey systems with a Holling type II. By reformulating this system as a set of two second-order differential equations, we investigate the nonlinear dynamics of the system from the Jacobi stability point of view using the Kosambi–Cartan–Chern (KCC) geometric theory. We then determine the nonlinear connection, the Berwald connection, and the five KCC invariants which express the intrinsic geometric properties of the system, including the deviation curvature tensor. Furthermore, we obtain the necessary and sufficient conditions for the parameters of the system in order to have the Jacobi stability near the equilibrium points, and we point these out on a few illustrative examples. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and analyze several new classes of multi-dimensional almost automorphic functions which generalize the classical one of Bochner. We develop the basic theory for the introduced classes, investigating the themes like composition principles, convolution invariance and the invariance under the actions of convolution products. We present several illustrative examples and applications to the abstract Volterra integro-differential equations and partial differential equations, providing also a mini appendix about almost automorphic functions on semi-topological groups. [ABSTRACT FROM AUTHOR]

Sonine kernel is characterized by the Sonine condition (denoted by SC) and is an important class of kernels in nonlocal differential equations and integral equations. This work proposes a SC with a more general form (denoted by gSC), which is more convenient than SC to accommodate complex kernels and equations. A typical kernel is given, and the first-kind Volterra integral equation under gSC is accordingly transformed and then analyzed. Based on these results, it is finally proved that the gSC is indeed equivalent to the original SC, which indicates that the Sonine kernel may be essentially characterized by the behavior of its convolution with the associated kernel at the starting point. [ABSTRACT FROM AUTHOR]

• A novel R package for modelling ecological dynamics was developed. • A three-cycle procedure was proposed to analyse a model and decide its complexity. • Graphical, analytical, and numerical tools were integrated to view model dynamics. Population dynamics modelling plays a crucial role in understanding ecological populations and making informed decisions for environmental management. However, existing software packages for dynamical system modelling often lack comprehensive integration of techniques and guidelines, limiting their practical usability. This paper introduces ecode, a novel package for modelling ecological populations and communities using ordinary differential equation systems, designed with a user-friendly framework. By following a three-cycle procedure, users can easily construct ecological models and explore their behaviours through a wide range of graphical, analytical, and numerical techniques. The package incorporates advanced techniques such as grid search methods and simulated annealing algorithms, enabling users to iteratively refine their models and achieve accurate predictions. Notably, ecode minimises external dependencies, ensuring robustness and reducing the risk of package failure caused by updates in dependencies. Overall, ecode serves as a valuable tool for ecological modelling, facilitating the exploration of complex ecological systems and the generation of informed predictions and management recommendations. [ABSTRACT FROM AUTHOR]

A method of writing the Fredholm or Volterra integral equation of the first kind is proposed aimed at finding the functional form of an arbitrary coefficient of the transfer equation. The described method is based on the use of functional analogs of the Hermit interpolation polynomial for approximate description of solutions of the transfer equations. The efficiency of the proposed method is demonstrated for the model problem of the stationary diffusion equation. [ABSTRACT FROM AUTHOR]

Crisis processes of biosystems often develop according to scenarios of rapid collapse, which cannot be predicted by statistical methods. This paper develops a method for constructing hybrid computational models using event-hierarchy representation for biophysical processes. System events calculated by the algorithm change the order of calculation of equations according to a given set of rules. The rules of physically conditioned redefinition of the right-hand sides of differential equations are used. The predicates employ the calculations of a group of accompanying characteristics, which are an integral part of the controlled biophysical dynamics. The model is investigated by presenting a computational scenario with a set of parameters, initial values, and an algorithm for making decisions about changing the impact for discrete time. Using computational experiments, a real outcome scenario for a situation that leads to the collapse of a biophysical system at a controlled level of exposure is described. The scenario sets the logic for making control decisions to change the level of external pressure on the natural environment. It is shown as a result of modeling that the transition of the process to an oscillatory mode leads to the choice of a risky control mode. It has been established that the dynamics of many real water populations has a point of threshold reduction in the efficiency of replenishment of stocks. The model scenario uses transformations of the phase portrait of iterations, which, in the presence of disconnected boundaries of the areas of attraction of alternative attractors and a strange chaotic repeller, lead to the uncertainty effects arising due to chaotic modes in a deterministic model. The properties of the described model scenario with a chaotic regime of dynamics are confirmed by the example of the catch of oceanic species of crustaceans. [ABSTRACT FROM AUTHOR]

This study is the first to use Laplace transform methods to solve a system of Caputo fractional Volterra integro-differential equations with variable coefficients and a constant multi-time delay. This technique is based on different types of kernels, which we will explain in this paper. Symmetry kernels, which have properties of difference kernels or simple degenerate kernels, are able to compute analytical work. These are demonstrated by solving certain examples and analyzing the effectiveness and precision of cause techniques. [ABSTRACT FROM AUTHOR]

Without taking into account external friction, small transverse vibrations of a viscoelastic pipeline of unit length are described for nonnegative values of time in dimensionless variables by an integro-differential equation with hinged conditions at the ends and with initial conditions. The solution of this equation can be written in terms of an operator semigroup. In the present paper, we establish that this equation generates a semigroup that is analytic in some sector of the right half-plane. [ABSTRACT FROM AUTHOR]

This article presents a numerical approach for solving the second kind of Volterra integro- differential equation (VIDE). The multistep block-Boole's rule method will estimate the solutions for the linear and nonlinear problems of VIDE. The method computes two solutions for VIDE along the interval. The proposed method is developed by derivation of the Lagrange interpolating polynomial. The convergence and stability analysis of the derived method are discussed. From the perspective of total function calls and time-saving, the computation results explained that the derived method performs better than other existing methods. [ABSTRACT FROM AUTHOR]

In this paper, we establish an approximate fixed point theorem for Γ-τ-sequentially continuous countably Φ τ -condensing mappings. As application, we obtain an approximate fixed point result for demicontinuous mappings and use this to prove new results in asymptotic fixed point theory. Moreover, we obtain new fixed point results for some countably Φ τ -condensing mappings and multivalued mappings and fixed point results of Krasnoselskii-Daher type for the sum of two τ-sequentially continuous mappings in non-separable Hausdorff topological vector spaces. We also present a multivalued version of an approximation result of Ky Fan for τ-Γ-s.l.sc multivalued mappings. Apart from that, we show the applicability of our results to the theory of Volterra integral equations in Banach spaces and we prove the existence of limiting-weak solutions for differential equations in Banach spaces not necessarily reflexive. Our results extend the results of Banaś and Ben Amar, Barroso and Seda. [ABSTRACT FROM AUTHOR]

The well-known central finite difference approximation was combined with the trapezoid quadrature method in this study to provide a numerical solution of the linear system of Volterra integro-fractional differential equations (LSVI-FDEs) of arbitrary orders, where the fractional derivative is described in the Caputo sense and the orders are between zero and one. The method works by first using the central finite difference approximation to approximate the Caputo derivative at any fixed point and then using the trapezoidal rule to obtain a finite difference expression for our fractional equation, while recalling the linear spline approximation for the first steps. This new, more efficient method involves converting sets of equations and conditions into matrix relationships, from which symmetry matrices can be created in some cases. We also present a new approach for error analysis of the discrete numerical scheme and the explicit numerical technique for LSVI-FDEs. The multi-level explicit finite difference approximation's stability and convergence were explored, and a MatLab application was created to explain the results. Finally, several numerical examples are offered to demonstrate the technique's application. [ABSTRACT FROM AUTHOR]

In this paper, we relate the notions of remote almost periodicity and quasi-asymptotical almost periodicity; in actual fact, we observe that a remotely almost periodic function is nothing else but a bounded, uniformly continuous quasi-asymptotically almost periodic function. We introduce and analyze several new classes of remotely c-almost periodic functions in ℝn, slowly oscillating functions in ℝn, and further analyze the recently introduced class of quasi-asymptotically c-almost periodic functions in ℝn. We provide certain applications of our theoretical results to the abstract Volterra integro-differential equations and the ordinary differential equations. [ABSTRACT FROM AUTHOR]

• The normal form formula of double-Hopf bifurcation is generated for general differential equations with the nonlocality. • The formula is applicable to functional differential equations and partial differential equations with or without the nonlocality. • The algorithm is favourable to reveal complex dynamics, e.g. periodic, quasi-periodic and multi-periodic oscillations. • Stable spatially nonhomogeneous periodic and quasi-periodic solutions are found in a nonlocal predator-prey model. This paper is concerned with the calculation of the normal form on the center manifold up to the third-order term at the double-Hopf singularity for general partial functional differential equations with the nonlocal effect. We derive explicit formulas of the normal form, that can be applied for both functional differential equations and partial differential equations with or without nonlocal effects in a bounded spatial domain. This provides an effective tool to establish existences of multi-periodic and quasi-periodic oscillations of a double-Hopf singularity for such equations. As an example, the Holling-Tanner predator-prey model with the nonlocal intraspecific competition of prey is considered. It turns out that double-Hopf bifurcation will occur because of the nonlocal interaction. Many spatio-temporal dynamics are found, including stable spatially homogeneous or nonhomogeneous periodic solutions, and stable spatially nonhomogeneous quasi-periodic solutions. [ABSTRACT FROM AUTHOR]

In this work, we present sufficient conditions to determine if the limit cycles of certain differential systems in the plane are algebraic or not. In particular, we obtain criteria such that the limit cycles of equations derived from predatory prey models with rational functional response are necessarily transcendental ovals. [ABSTRACT FROM AUTHOR]

In this research work, the existence and global exponential stability of the positive periodic solutions of Nicholson's blowflies equation with multiple different time-varying delays are investigated. Without adopting the restriction assumption on the maximum rate of reproduction function, we establish a criterion to ensure the existence and global exponential stability of the positive periodic solutions for the addressed equation by using differential inequality techniques, the fluctuation lemma and Lyapunov functional method. The obtained result is new, which improves and complements some existing ones. Moreover, the effectiveness of the theoretical finding is illustrated by an example with numerical simulation. [ABSTRACT FROM AUTHOR]