International Journal of Biomathematics

This paper is concerned with a system of semilinear parabolic equations with two free boundaries describing the spreading fronts of the invasive species in a mutualistic ecological model. The local existence and uniqueness of a classical solution are obtained and the asymptotic behavior of the free boundary problem is studied. Our results indicate that two free boundaries tend monotonically to finite or infinite limits at the same time, and the free boundary problem admits a global slow solution with unbounded free boundaries if the intra-specific competitions are strong, while if the intra-specific competitions are weak, there exist the blowup solution and global fast solution.

In this paper, we study the fractional-order biological population models (FBPMs) with Malthusian, Verhulst, and porous media laws. The fractional derivative is defined in Caputo sense. The optimal homotopy asymptotic method (OHAM) for partial differential equations (PDEs) is extended and successfully implemented to solve FBPMs. Third-order approximate solutions are obtained and compared with the exact solutions. The numerical results unveil that the proposed extension in the OHAM for fractional-order differential problems is very effective and simple in computation. The results reveal the effectiveness with high accuracy and extremely efficient to handle most complicated biological population models.

In this paper, an SEIR epidemic model with vaccination is formulated. The results of our mathematical analysis indicate that the basic reproduction number plays an important role in studying the dynamics of the system. If the basic reproduction number is less than unity, it is shown that the disease-free equilibrium is globally asymptotically stable by comparison arguments. If it is greater than unity, the system is permanent and there is a unique endemic equilibrium. In this case, sufficient conditions are established to guarantee the global stability of the endemic equilibrium by the theory of the compound matrices. Numerical simulations are presented to illustrate the main results.

In this paper, the dynamics of a stochastic model for algal bloom with nutrient recycling is investigated. The model incorporates a white noise term in the growth equation of algae population to describe the effects of random fluctuations in the environment, and a nutrient recycling term in the nutrient equation to account for the conversion of detritus into nutrient. The existence and uniqueness of the global positive solution of the model is first proved. Then we study the long-time behavior of the model. Conditions for the extinction and persistence in the mean of the algae population are established. By using the theory of integral Markov semigroups, we show that the model has an invariant and asymptotically stable density. Numerical simulations illustrate our theoretical results.

The pattern formation in reaction–diffusion system has long been the subject of interest to the researchers in the domain of mathematical ecology because of its universal existence and importance. The present investigation deals with a spatial dynamics of the Beddington–DeAngelis predator–prey model in the presence of a constant proportion of prey refuge. The model system representing boundary value problem under study is subjected to homogeneous Neumann boundary conditions. The asymptotic stability including the local and the global stability and the bifurcation as well of the unique positive homogeneous steady state of the corresponding temporal model has been analyzed. The Turing instability region in two-parameter space and the condition of diffusion-driven instability of the spatiotemporal model are investigated. Based on the appropriate numerical simulations, the present model dynamics in Turing space appears to get influenced by prey refuge while it exhibits diffusion-controlled pattern formation growth to spots, stripe-spot mixtures, labyrinthine, stripe-hole mixtures and holes replication. The results obtained appear to enrich the findings of the model system under consideration.

A mathematical model describing the dynamics of toxin producing phytoplankton–zooplankton interaction with instantaneous nutrient recycling is proposed. We have explored the dynamics of plankton ecosystem with multiple delays; one due to gestation period in the growth of phytoplankton population and second due to the delay in toxin liberated by TPP. It is established that a sequence of Hopf bifurcations occurs at the interior equilibrium as the delay increases through its critical value. The direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined using the theory of normal form and center manifold. Meanwhile, effect of toxin on the stability of delayed plankton system is also established numerically. Finally, numerical simulations are carried out to support and supplement the analytical findings.

This work represents a formalized description of information and communicative interactions of individuals on the basis of theory of information images (II). It also demonstrates how important it is to choose the models type adequate to the systems under research. It also introduces an explication of the possibility to create the model of information and communication interactions that can illustrate transmission of information between two and more individuals. Methods and approaches suggested in this paper allow us to compare different levels of the described processes depending on the chosen architecture of the model; the methods and approaches mentioned in the current work can be also used to simulate the processes of distortion and generation of information images while information and communication social interaction. Expansion and addition of information images theory in terms of transmission of information among individuals enables us to speak about the space of individual II. The existence of such a space and creation of correct formalized model help us to explain a number of characteristic phenomena of human thinking processes. As a result of this research, the authors introduce an equation that describes the spatial and structural evolution of individual II. There is also an example of modeling on the basis of this theory taking into account the results of the experiment (bilingual Stroop test).

The paper analyzes the influence of a susceptible–infectious–susceptible (SIS) infectious disease affecting both fish and broiler species. The paper also considers a joint SIS project of fish and broiler in which the growth rates of both species vary with available nutrients and environmental carrying capacities of biomasses. The nutrients for both species are functions of the biomasses of the two species. The harvesting rates of fish and broiler depend linearly on common effort function. It is assumed that the diseases are transmitted to the susceptible populations by direct contact with the infected populations. Using the medicine, some portion of the infected populations are transmitted to the susceptible populations. The existence of steady states and their stability are investigated analytically. The joint profit of the SIS model is maximized using Pontryagin’s maximum principle and corresponding optimum harvesting rates are also obtained. Using Mathematica software, the models are illustrated and the optimum results are obtained and presented in tabular and graphical forms.

Understanding of ecosystem resilience and stability requires comprehending predator–prey dynamics because ecosystems consist of dynamically interacting subsystems that include predator–prey relationships. This relationship is closely related to the hunting–escaping strategies employed by the predator and prey. Therefore, understanding the effects of hunting and escaping strategies on ecosystems will lead to a better understanding of these systems. As an approach for describing the predator–prey interaction, lattice-based models have been adopted because this approach has strong advantages for simulating various dynamical processes of individual–individual interaction. In the models, each lattice cell is either considered as an attractive/repulsive cell, or an individual cell, or else it is empty. The attractive (or repulsive cell) can be interpreted as the prey (or predator) of the individual. These states allow us to incorporate the ecological processes of local antagonistic interactions, namely the spread of disturbances (by the predator) and regrowth or recovery (by the prey). These processes are directly related to the strategic behavior of individuals, such as hunting and escaping. In this study, we suggest a simple and effective mapping formula as a stochastic rule to describe the hunting and escaping behavior. This formula could be widely used not only in the behavior but also in competitive and cooperative relationships.

In this paper, analytical result of avian–human influenza epidemic model has been investigated by applying homotopy analysis method (HAM) and by expanding it to hybrid numeric-analytic method which is known as multistage HAM (MSHAM). HAM is an algorithm which gives us the approximate solution of the problem in an arrangement of time interims and by modifying it to multistage one. Some advantages such as flexibility of picking the auxiliary linear operator and the auxiliary parameter are emerged, that leads us to achieve some excellent results in this work. Furthermore, in this analytical work, obtained results are compared and reported with numerical ones which were obtained previously from methods such as the Runge–Kutta (RK4) method.

This paper presents a novel analysis for the solution of nonlinear age-structured problem which is of extreme importance in biological sciences. The presented model is very useful but quite complicated. Modified variational iteration method (MVIM) coupled with auxiliary parameter is used to cope with the complexity of the model which subsequently shows better results as compared to some existing results available in literature. Furthermore, an appropriate way is used for the identification of auxiliary parameter by means of residual function. Numerical examples are presented for the analysis of the proposed algorithm. Graphical results along with the discussions re-confirm the efficiency of proposed algorithm. The work proposes a new algorithm where He’s polynomials and an auxiliary parameter are merged with correction functional. The suggested scheme is implemented on nonlinear age-structured population models. Graphs are plotted for the residual function that reflects the accuracy and convergence of the presented algorithm.

In this paper, we study the qualitative behavior of a discrete-time epidemic model. More precisely, we investigate equilibrium points, asymptotic stability of both disease-free equilibrium and the endemic equilibrium. Furthermore, by using comparison method, we obtain the global stability of these equilibrium points under certain parametric conditions. Some illustrative examples are provided to support our theoretical discussion.

We investigate the stability of steady states of a size- and stage-structured population model, which is a hybrid system of ordinary and partial differential equations with global integral feedbacks. After the formulation of a criterion by spectrum method, we derive conditions for global stability of the trivial state and local stability of the positive equilibrium via the basic reproduction rate. Furthermore, some examples and simulations are presented to illustrate the obtained results.

The surface electromyography (SEMG) is a complicated biomedical signal, generated during voluntary or involuntary muscle activities and these muscle activities are always controlled by the nervous system. In this paper, the processing and analysis of SEMG signals at multiple muscle points for different operations were carried out. Myoelectric signals were detected using designed acquisition setup which consists of an instrumentation amplifier, filter circuit, an amplifier with gain adjustment. Further, Labview${}^{Ⓡ}$-based data programming code was used to record SEMG signals for independent activities. The whole system consists of bipolar noninvasive electrodes, signal acquisition protocols and signal conditioning at different levels. This work uses recorded SEMG signals generated by biceps and triceps muscles for four different arm activities. Feature extraction was done on the recorded signal for investigating the voluntary muscular contraction relationship for exercising statistic measured index method to evaluate distance between two independent groups by directly addressing the quality of signal in separability class for different arm movements. Thereafter repeated factorial analysis of variance technique was implemented to evaluate the effectiveness of processed signal. From these results, it demonstrates that the proposed method can be used as SEMG feature evaluation index.

Several biological processes, such as convective nutrient transport and convective drug delivery in biological tissues involves the transvascular and interstitial movement of biofluids. This work addresses transvascular and interstitial transport of nutrient inside a spherical tumor. Most of the biological tissues behave like deformable porous material and show mechanical behavior towards the fluid motion, due to the fact, that the forces like the drag, which is associated with fluid flow may compress the tissue material. On the macroscopic level, transport of solutes like nutrients, drug molecules, etc. within the tumor interstitial space is modeled. The hydrodynamic problem is treated with biphasic mixture theory under steady state and spherically symmetry situation. The transvascular transport of nutrient is modeled with the modified Sterling’s equation. The present model describes the overall nutrient distribution and predicts various criteria for the necrosis formation inside the tumor. Present study justifies that the parameters, which controls the nutrient supply to the tumor interstitial space through the blood vessel network inside the tumor, competes with reversible nutrient consumption kinetics of the tumor cells. This study also finds the role of some of those parameters on the deformation of cellular phase of the tumor as a consequence of interstitial fluid flow.

United Nations Political Declaration 2011 on HIV and AIDS calls to reduce the sexual transmission and the transmission of HIV among people, who inject drugs by $50\%$ by 2015, through different control strategies and precautionary measures. In this paper, we propose and study a simple SI type model that considers the effect of various precautionary measures to control HIV epidemic. We show, unlike conventional epidemic models, that the basic reproduction number which essentially considered as the disease eradication condition is no longer sufficient to eliminate HIV infection. In particular, we show that even when the basic reproduction number is made less than unity, the disease may persist if the initial outbreak is not low. Eradication of disease is however guaranteed if the ensemble control measure exceeds some upper critical value. It is also shown that an epidemic model with mass action incidence may exhibit backward bifurcation and bistability if density-dependent demography is considered. Our theoretical study thus indicates that extra attention should be given in controlling HIV epidemic to achieve the desired result.