Logistic Growth Term Research Articles

Discontinuity waves in temperature and diffusion models

We analyse shock wave behaviour in a hyperbolic diffusion system with a general forcing term which is qualitatively not dissimilar to a logistic growth term. The amplitude behaviour is interesting and depends critically on a parameter in the forcing term. We also develop a fully nonlinear acceleration wave analysis for a hyperbolic theory of diffusion coupled to temperature evolution. We consider a rigid body and we show that for three-dimensional waves there is a fast wave and a slow wave. The amplitude equation is derived exactly for a one-dimensional (plane) wave and the amplitude is found for a wave moving into a region of constant temperature and solute concentration. This analysis is generalized to allow for forcing terms of Selkov–Schnakenberg, or Al Ghoul-Eu cubic reaction type. We briefly consider a nonlinear acceleration wave in a heat conduction theory with two solutes present, resulting in a model with equations for temperature and each of two solute concentrations. Here it is shown that three waves may propagate.

On the effects of density-dependent emigration on ecological models with logistic and weak Allee type growth terms

On the effects of density-dependent emigration on ecological models with logistic and weak Allee type growth terms

On the existence of solutions for a parabolic‐elliptic chemotaxis model with flux limitation and logistic source

In this article, we study the existence of solutions of a parabolic‐elliptic system of partial differential equations describing the behaviour of a biological species “ ” and a chemical stimulus “ ” in a bounded and regular domain of . The equation for is a parabolic equation with a nonlinear second order term of chemotaxis type with flux limitation as for . The chemical substance distribution satisfies the elliptic equation The evolution of is also determined by a logistic type growth term . The system is studied under homogeneous Neumann boundary conditions. The main result of the article is the existence of uniformly bounded solutions for and any .

Viral Infection Model with Diffusion and Distributed Delay: Finite-Dimensional Global Attractor.

We study a virus dynamics model with reaction-diffusion, logistic growth terms and a general non-linear infection rate functional response. The model has a distributed delay, including the case of state-selective delay. We construct a dynamical system in a Hilbert space and prove the existence of a finite-dimensional global attractor.

A Continuum Mathematical Model of Substrate-Mediated Tissue Growth

We consider a continuum mathematical model of biological tissue formation inspired by recent experiments describing thin tissue growth in 3D-printed bioscaffolds. The continuum model, which we call the substrate model, involves a partial differential equation describing the density of tissue, {hat{u}}(hat{{mathbf {x}}},{hat{t}}) that is coupled to the concentration of an immobile extracellular substrate, {hat{s}}(hat{{mathbf {x}}},{hat{t}}). Cell migration is modelled with a nonlinear diffusion term, where the diffusive flux is proportional to {hat{s}}, while a logistic growth term models cell proliferation. The extracellular substrate {hat{s}} is produced by cells and undergoes linear decay. Preliminary numerical simulations show that this mathematical model is able to recapitulate key features of recent tissue growth experiments, including the formation of sharp fronts. To provide a deeper understanding of the model we analyse travelling wave solutions of the substrate model, showing that the model supports both sharp-fronted travelling wave solutions that move with a minimum wave speed, c = c_{mathrm{min}}, as well as smooth-fronted travelling wave solutions that move with a faster travelling wave speed, c > c_{mathrm{min}}. We provide a geometric interpretation that explains the difference between smooth and sharp-fronted travelling wave solutions that is based on a slow manifold reduction of the desingularised three-dimensional phase space. In addition, we also develop and test a series of useful approximations that describe the shape of the travelling wave solutions in various limits. These approximations apply to both the sharp-fronted and smooth-fronted travelling wave solutions. Software to implement all calculations is available at GitHub.

Traveling waves, blow‐up, and extinction in the Fisher–Stefan model

AbstractWhile there is a long history of employing moving boundary problems in physics, in particular via Stefan problems for heat conduction accompanied by a change of phase, more recently such approaches have been adapted to study biological invasion. For example, when a logistic growth term is added to the governing partial differential equation in a Stefan problem, one arrives at the Fisher–Stefan model, a generalization of the well‐known Fisher–KPP model, characterized by a leakage coefficient which relates the speed of the moving boundary to the flux of population there. This Fisher–Stefan model overcomes one of the well‐known limitations of the Fisher–KPP model, since time‐dependent solutions of the Fisher–Stefan model involve a well‐defined front which is more natural in terms of mathematical modeling. Almost all of the existing analysis of the standard Fisher–Stefan model involves setting , which can lead to either invading traveling wave solutions or complete extinction of the population. Here, we demonstrate how setting leads to retreating traveling waves and an interesting transition to finite‐time blow‐up. For certain initial conditions, population extinction is also observed. Our approach involves studying time‐dependent solutions of the governing equations, phase plane, and asymptotic analysis, leading to new insight into the possibilities of traveling waves, blow‐up, and extinction for this moving boundary problem. MATLAB software used to generate the results in this work is available on Github.

Stability and bifurcation analysis of hepatitis B‐type virus infection model

The main aim of this study is to analyze the dynamical properties of hepatitis B‐type virus (HBV) infection in terms of mathematical model. The presented mathematical model on HBV involves the various factors such as immune impairment, total carrying capacity, logistic growth term, and antiretroviral therapies. In addition, the effect of time delays is also considered into the model, which are inevitable during the activation of immune response and time taken to infect the healthy cells. Mathematically, the qualitative analyses such as stability, bifurcation, and stabilization analysis are performed to explore the dynamical characteristics of HBV over the period of time. The significance of the model parameters is revealed through Hopf‐type bifurcation analysis and the global stability analysis of the proposed model. With the help of data set values that are extracted from the literature, the efficiency of the derived theoretical results is explored.

Dynamical analysis and effects of law enforcement in a social interaction model

Crime is a rising problem all over the world. Over the years, several researches have been conducted to coin suitable model systems and intervention strategies that would decrease delinquent behavior and promote prosocial development. Being dynamic and complex process, the spread of crime requires a system level approach. In this paper, we formulate and analyze a series of dynamical model systems of the dilation of crime incorporating law enforcement. The population models of social interactions have been constructed based on predator–prey interaction models with Holling type II response function. Two different types of populations (criminal minded and non-criminal minded) have been assumed in the given community/society. According to law policy for crime control, we scrutinize the dynamic behavior of the model system concerning law enforcement on the criminal minded population. Along with analytical expressions for the existence of different equilibrium points and their stability, we have also provided their geometrical interpretations using isocline analysis. The expressions obtained for the existence and stability of equilibrium points have been used to investigate the effects of coefficient of law enforcement and the logistic growth term on the prevalence of crime. It is observed that for a threshold value of law enforcement, a stable limit cycle exists. In particular, a threshold of law enforcement is determined beyond which the associated place/community could be made crime free. A threshold value R0 (similar to basic reproduction number in epidemiology) has also been introduced. It is shown that when R0<1, crime free equilibrium is stable. The elimination and persistence of crime have been discussed via two parameters (law enforcement and half saturation constant) bifurcation diagram. Theoretical results have been supported via numerical simulations followed by discussion and societal impacts of the present study.

Mathematical models for diseases in wildlife populations with indirect transmission

In this paper, five different models for five different kinds of diseases occurring in wildlife populations are introduced. In all models, a logistic growth term is taken into account and the disease is transmitted to the susceptible population indirectly through an environment reservoir. The time evolution of these diseases is described together with its spatial propagation. The character of spatial homogeneous equilibria against the uniform and non-uniform perturbations together with the occurrence of Hopf bifurcations are discussed through a linear stability analysis. No Turing instability is observed. The partial differential field equations are also integrated numerically to validate the stability results herein obtained and to extract additional information on the temporal and spatial behavior of the different diseases.

On dynamics of a mathematical model for HIV infection with fusion effect and cure rate

In this manuscript, we have proposed and analyzed a differential equation model for HIV infection to study the dynamics of three different populations: HIV-free CD4 + T cells, HIV-infected CD4 + T cells and free virus of the model. In the model, we have incorporated fusion effect for HIV-free CD4 + T cell and free virus, proliferation of HIV-free CD4 + T cells which follows a full logistic growth term and cure rate for HIV-infected CD4 + T cells. Our main objective is to investigate the effects of fusion and cure rate on the dynamics of the model. We have used next generation matrix method to calculate the basic reproduction number (R 0 ) for this proposed model. Local stability of the existing equilibrium points is discussed using Routh-Hurwitz theorem. Also, in order to establish the global stability criteria Lyapunov functional and geometric approach are used. From the analysis it is found that if the basic reproduction number R 0 ≤ 1, HIV will be removed from the population of CD4 + T cells and if R 0 &gt; 1, there exists chronic infection. Also, we have carried out numerical simulations in order to verify the analytic results.

Phase transitions in a two-species model for cell segregation and logistic growth

We study a model of cell segregation in a population composed of two cell types. Starting from a model initially proposed in [3], we aim to understand the impact of a cell division process on the system’s segregation abilities. The original model describes a population of spherical cells interacting with their close neighbors by means of a repulsion potential and which centers are subject to Brownian motion. Here, we add a stochastic birth-death process in the agent-based model, that approaches a logistic growth term in the continuum limit. We address the linear stability of the spatially homogeneous steady states of the macroscopic model and obtain a precise criterion for the phase transition, which links the system segregation ability to the model parameters. By comparing the criterion with the one obtained without logistic growth, we show that the system’s segregation ability is the result of a complex interplay between logistic growth, diffusion and mechanical repulsive interactions. Numerical simulations are presented to illustrate the results obtained at the microscopic scale.

On the global generalized solvability of a chemotaxis model with signal absorption and logistic growth terms

Introducing a suitable solution concept, we show that in bounded smooth domains , , the initial boundary value problem for the chemotaxis system with homogeneous Neumann boundary conditions and widely arbitrary initial data has a generalized global solution for any .

Explicit impacts of harvesting in delayed predator-prey models

Abstract Modeling population dynamics using delay differential equations and exploring the impacts of harvesting in predator-prey systems are among few of the thrust areas of research in theoretical and applied ecology. Many results are established to understand distinct dynamics under population harvesting. However, comparatively less attention is paid to explain the explicit harvesting effects when populations fluctuate due to time delay. In this contribution, two well known Lotka–Volterra (LV) type and Rosenzweig–MacArthur (RM) predator-prey models incorporating time delay into the logistic growth term are considered. The analysis and the obtained results are summarized as follows. (a) Firstly, the dynamics of both the models, by considering the time delay as the bifurcation parameter, is analyzed. Some of the parameter conditions for the delay induced stability switching are improved and corrected in comparison to the earlier works. The delay induced stability results are derived and found to be similar for both the models. (b) We investigate whether harvesting of either prey or predator can locally stabilize (respectively, destabilize) the system when the unharvested system dynamics is at non-equilibrium (respectively, stable steady state) mode due to time delay. It is observed that harvesting can induce stability and instability switching depending upon the dynamics mode of the unharvested system. (c) In the same framework, we examine if a stable steady state can be obtained when predator is harvested towards Maximum Sustainable Yield (MSY) level. Unlike the case of non-delayed LV type and RM predator-prey models, it is found that harvesting the predator towards MSY in the delayed models does not guarantee a stable stock. The new results compared to the existing literature might contribute in enriching fishery management policy and theoretical ecology as a whole.

Stability Analysis of Delayed Immune Response BCG Infection in Bladder Cancer Treatment Model by Stochastic Perturbations.

We present a revised mathematical model of the immune response to Bacillus Calmette-Guérin (BCG) treatment of bladder cancer, optimized according to biological and clinical data accumulated during the last decade. The improved model accounts for cytotoxic T lymphocyte differentiation as an integral element of the delayed immune response, as well as the logistic growth terms for cancer cell proliferation. Three equilibria are demonstrated for the proposed model, which is assumed to be influenced by white noise stochastic perturbations that are directly proportional to the system state deviation from an equilibrium. Stability conditions for all equilibria are analyzed using the Kolmanovskii-Shaikhet general method of Lyapunov functionals construction.

Bifurcation structure of stationary solutions for a chemotaxis system with bistable growth

From the viewpoint of pattern formation, Keller–Segel systems with growth terms are studied. These models exhibit various stationary and spatio-temporal patterns which are caused by a combination of three effects: chemotaxis, diffusion and growth. In this paper, we consider Keller–Segel system with the cubic growth term known as the Allee effect in ecology and its shadow system in the limiting case that the mobility of biological population tends to infinity. We show the existence and stability of stationary solutions of the shadow system in one space dimension. Our proof is based on the bifurcation theory, a singular perturbation method and a level set analysis. We also show some numerical results on global structures of stationary solutions in the systems by using AUTO package. Moreover, we mention the difference in dynamics between Keller–Segel system with the cubic growth term and that with the logistic growth term with the aid of a computer.

Modeling the role of acquired immune response and antiretroviral therapy in the dynamics of HIV infection

This paper deals with the study of a virus dynamics model in order to get better insights into HIV infection within the body. The model incorporates therapeutic modalities such as reverse transcriptase inhibitors (RTIs) and protease inhibitors (PIs). RTIs prevent viral replication/entry within the infected CD4+ T cells while PIs block the virus assembly and thus further propagation and production of new virions. The proliferation of uninfected CD4+ T cells has been assumed to be as full logistic growth term to capture the dynamics of HIV virus. The model also considers two important components of the acquired immune response, namely the cytotoxic T lymphocyte (CTL) immune response (self stimulation due to infection and stimulation due to infected cells have been considered) and antibody immune response. Critical threshold conditions for the existence of equilibrium points have been determined. We studied the analytical behavior of these equilibrium points locally as well as globally using Lasalle’s invariance principle and Lyapunov’s direct method. We explored the sensitivity of the therapeutic drugs on the model system. Further, the behavior of the proposed model system has been studied numerically through simulation tools.

Large time periodic solution to the coupled chemotaxis‐Stokes model

In this paper, we deal with the time periodic problem for the coupled chemotaxis‐fluid model with logistic growth term. We prove the existence of large time periodic solution in spatial dimension . Furthermore, we also show that if the time periodic source g and the potential force belong to , the solution is also a classical solution.

On the stability of homogeneous steady states of a chemotaxis system with logistic growth term

We consider a nonlinear PDEs system of Parabolic–Elliptic type with chemotactic terms. The system models the movement of a population “ n ” towards a higher concentration of a chemical “ c ” in a bounded domain Ω . We consider constant chemotactic sensitivity χ and an elliptic equation to describe the distribution of the chemical n t − d n Δ n = − χ div ( n ∇ c ) + μ n ( 1 − n ) , − d c Δ c + c = h ( n ) for a monotone increasing and Lipschitz function h . We study the asymptotic behavior of solutions under the assumption of 2 χ | h ′ | < μ . As a result of the asymptotic stability we obtain the uniqueness of the strictly positive steady states.

Stability and multi-parametric Hopf bifurcation analyses of viral infection model with time delay

Ever since HIV was first diagnosed in human, a great number of scientific works have been undertaken to explore the biological mechanisms involved in the infection and progression of the disease. This paper deals with stability and bifurcation analyses of mathematical model that represents the dynamics of HIV infection of thymus. The existence and stability of the equilibria are investigated. The model is described by a system of delay differential equations with logistic growth term, cure rate and discrete type of time delay. Choosing the time delay as a bifurcation parameter, the analysis is mainly focused on the Hopf bifurcation problem to predict the existence of a limit cycle bifurcating from the infected steady state. Further, using center manifold theory and normal form method we derive explicit formulae to determine the stability and direction of the limit cycles. Moreover the mitosis rate r also plays a vital role in the model, so we fix it as second bifurcation parameter in the incidence of viral infection. Our analysis shows that, while both the bifurcation parameters can destabilize the equilibrium E* and cause limit cycles. Numerical simulations are performed to investigate the qualitative behaviors of the inherent model.

Sensitivity and uncertainty analysis of a simplified Kirschner-Panetta model for immunotherapy of tumor-immune interaction

In this study, we have simplified the Kirschner-Panetta model on the interaction of tumor cells and effector cells by considering linear growth term as opposed to logistic growth term used by Kirschner and Panetta. We have done a comprehensive mathematical analysis and established the existence of positive equilibrium. In addition, a fixed point bifurcation is investigated using the rate of spread of tumor as a varying parameter, suggesting that backward bifurcation can occur under reasonable choice of parameters. However, bistable dynamics is unlikely to happen in this case, which implies that the strategies that could reduce the rate of spread of tumor are the most influential to cancer treatment. Through mathematical deduction and numerical simulation, an elaborate uncertainty and sensitivity analysis of the rate of spread of tumor $R_{s}$ is performed. The distribution of $R_{s}$ is derived, and the sensitivity of the magnitude of $R_{s}$ to the uncertainty in estimating values of input parameters is assessed. The results indicate that the external source of effector cells and its death rate are influential in the rate of spread of tumor.