Dulac Function Research Articles

The Impact of Predator-dependent Prey Refuge on the Dynamics of a Leslie-Gower Predator-prey Model

In this paper, we propose a new Leslie-Gower predator-prey model with predator-dependent prey refuge. Firstly, we obtain the positivity and boundedness of the system solution. Secondly, we prove that the origin is unstable using blow-up method, analyze the existence and local stability of the boundary equilibrium point and positive equilibrium point, and prove that the unique positive equilibrium point of the system is globally asymptotically stable by constructing a suitable Dulac function. Finally, mathematic analysis and numerical simulation show that: (1) when the strength of the predator-dependent prey refuge k = 0 , the dynamics of the predator-prey system without predator-dependent prey refuge are consistent with the results obtained from the traditional Leslie-Gower predator-prey system; (2) when k tends to positive infinity, the predator-dependent refuge lead to prey population densities fall somewhere between without prey refuge and with proportional refuge. However, the predator densities within this new form of the predator-dependent prey refuge is greater than the densities of predators without prey refuge and with proportional refuge; (3) increasing the strength k of the predator-dependent prey refuge can increase the densities of predator and prey populations respectively.

Global dynamics of a tumor-immune model with an immune checkpoint inhibitor

In this paper, a mathematical ordinary differential tumor-immune model is proposed based on an immune checkpoint inhibitor, which is an innovative method for tumor immunotherapies. Two important factors in tumor-immune response are the programmed cell death protein 1 (PD-1) and its ligand PD-L1. The model consists of three populations: tumor cells, activated T cells and anti-PD-1. By analyzing the dynamics of the model, it is found that there is always a unique tumor-free equilibrium and at most two tumor interior equilibria. The nonexistence of nontrivial positive periodic orbits is established by using the new Dulac function, and then a global dynamics of the model is obtained. The conclusions of our analysis show that increasing the possibility of T cells killing tumor cells ([Formula: see text]), early detection of tumor cells, or the use of PD-1 inhibitors to activate T cells are effective in eliminating tumor cells.

Dynamical properties of single species stage structured model with Michaelis-Menten type harvesting on adult population and linear harvesting on juvenile population

A very common and effective way for investigating future demographics is the study of stage structured models. The focus of this article is to propose a modified model to study the impact of population harvesting on their juvenile and adult stages, and analyze the dynamical properties from both qualitative and numerical perspective. It studies single species stage structured model with linear harvesting on juvenile group and Michaelis-Menten type harvesting on adult group. We exploit general ideas in mathematical modeling process to study the dynamical properties and their biological, ecological, and economic implications. It discusses that bi-stability phenomena may exist, global asymptotic stability at boundary equilibrium points and internal equilibrium points are investigated from construction of suitable Lyapunov and Dulac functions. It has been observed that a suitable linear harvesting on juvenile population can feasibly be carry out along with Michaelis-Menten type harvesting on adult population without endangering extinction of any group of population.

Dynamics Analysis of Cannibalistic Model With Density Dependence in Mature Stage

Based on considering mature individuals with density dependence, a two-stage structure model of egg-mature individuals with cannibalism is established. The dynamic behavior of the equilibrium of the model is discussed from two aspects: when no cannibalism exists, the global asymptotic stability of the equilibria is proved by the Lyapunov theorem; when cannibalism exists, it makes the model have saddle-node bifurcation by using the center manifold theorem. By constructing Dulac function, there is no limit cycle in the two-dimensional autonomous system, therefore, the global stability of the equilibria is obtained. Finally, the theoretical results are verified by numerical simulation.

Localizing branes with bifurcating bulks

We study the problem of evolution of bulk 5-fluids having an embedded braneworld with a flat, de Sitter, or anti-de Sitter geometry. We introduce new variables to express the Einstein equations as a dynamical system that depends on the equation of state parameter gamma and exponent lambda . For linear fluids (i.e., lambda =1), our formulation leads to a partial decoupling of the equations and thus to an exact solution. We find that such a fluid develops a transcritical bifurcation around the value gamma =-1/2, and study how this behaviour affects to stability of the solutions. For nonlinear fluids, the situation is more diverse. We find an overall attractor at lambda =1/2 and draw enough phase portraits to exhibit in detail the overall dynamics. We show that the value lambda =3/2 is structurally unstable and typical for other forms of lambda . Consequently, we observe a noticeable dependence of the qualitative behaviour of the solutions on different ‘polytropic’ forms of the fluid bulk. In addition, we prove the existence of a Dulac function for nonlinear fluids, signifying the impossibility of closed orbits in certain subsets of the phase space. We also provide ample numerical evidence of gravity localizing solutions on the brane which satisfy all energy conditions.

To study the transmission dynamic of SARS-CoV-2 using nonlinear saturated incidence rate

In this work, we construct a new SARS-CoV-2 mathematical model of SQIR type. The considered model has four compartments as susceptible S, quarantine Q, infected I and recovered R. Here saturated nonlinear incidence rate is used for the transmission of the disease. We formulate our model first and then the disease-free and endemic equilibrium (EE) are calculated. Further, the basic reproduction number is computed via the next generation matrix method. Also on using the idea of Dulac function, the global stability for the proposed model is discussed. By using the Routh–Hurwitz criteria, local stability is investigated. Through nonstandard finite difference (NSFD) scheme, numerical simulations are performed. Keeping in mind the significant importance of fractional calculus in recent time, the considered model is also investigated under fractional order derivative in Caputo sense. Finally, numerical interpretation of the model by using various fractional order derivatives are provided. For fractional order model, we utilize fractional order NSFD method. Comparison with some real data is also given.

Cosmological global dynamical systems analysis

We consider a dynamical systems formulation for models with an exponential scalar field and matter with a linear equation of state in a spatially flat and isotropic spacetime. In contrast to earlier work, which only considered linear hyperbolic fixed point analysis, we do a center manifold analysis of the non-hyperbolic fixed points associated with bifurcations. More importantly though, we construct monotonic functions and a Dulac function. Together with the complete local fixed point analysis this leads to proofs that describe the entire global dynamics of these models, thereby complementing previous local results in the literature.

Effectiveness of the Bendixson–Dulac theorem

We illustrate with several new applications the power and elegance of the Bendixson–Dulac theorem to obtain upper bounds of the number of limit cycles for several families of planar vector fields. In some cases we propose to use a function related with the curvature of the orbits of the vector field as a Dulac function. We get some general results for Liénard type equations and for rigid planar systems. We also present a remarkable phenomenon: for each integer m≥2, we provide a simple 1-parametric differential system for which we prove that it has limit cycles only for the values of the parameter in a subset of an interval of length smaller that 32(3/m)m/2 that decreases exponentially when m grows. One of the strengths of the results presented in this work is that although they are obtained with simple calculations, that can be easily checked by hand, they improve and extend previous studies. Another one is that, for certain systems, it is possible to reduce the question of the number of limit cycles to the study of the shape of a planar curve and the sign of an associated function in one or two variables.

Bifurcation analysis and global dynamics in a predator–prey system of Leslie type with an increasing functional response

The dynamical behaviors of a Leslie type predator–prey system are explored when the functional response is increasing for both predator and prey. Qualitative and quantitative analysis methods based on stability theory, bifurcation theory and numerical simulation are adopted. It is showed that the system is dissipative and permanent, and its solutions are bounded. Global stability of the unique positive equilibrium is investigated by constructing Dulac function and applying Poincaré–Bendixson theorem. The bifurcation behaviors are further explored and the number of limit cycles is determined. By calculating the first Lyapunov number and the first two focus values, it is proved that the positive equilibrium is not a center but a weak focus of multiplicity at most two, so the system undergoes Hopf bifurcation and Bautin bifurcation. The normal form of Bautin bifurcation is also obtained by introducing the complex system. Moreover, numerical simulations are run to demonstrate the validity of theoretical results.

Global analysis of a cancer model with drug resistance due to Lamarckian induction and microvesicle transfer

Development of resistance to chemotherapy in cancer patients strongly effects the outcome of the treatment. Due to chemotherapeutic agents, resistance can emerge by Darwinian evolution. Besides this, acquired drug resistance may arise via changes in gene expression. A recent discovery in cancer research uncovered a third possibility, indicating that this phenotype conversion can occur through the transfer of microvesicles from resistant to sensitive cells, a mechanism resembling the spread of an infectious agent. We present a model describing the evolution of sensitive and resistant tumour cells considering Darwinian selection, Lamarckian induction and microvesicle transfer. We identify three threshold parameters which determine the existence and stability of the three possible equilibria. Using a simple Dulac function, we give a complete description of the dynamics of the model depending on the three threshold parameters.We also establish an agent based model as a spatial version of the ODE model and compare the outputs of the two models. We find that although the ODE model does not provide spatial information about the structure of the tumour, it is capable to determine the outcome in terms of tumour size and distribution of cell types.We demonstrate the possible effects of increasing drug concentration, and characterize the possible bifurcation sequences. Our results show that the presence of microvesicle transfer cannot ruin a therapy that otherwise leads to extinction, however it may doom a partially successful therapy to failure.

Holling-Tanner Predator-Prey Model with Type-IV Functional Response and Harvesting

In this paper we modify Holling-Tanner predator-prey model by using type-IV functional response in prey species in lieu of type-II functional response. Harvesting is used in predator as well as prey species. This model is compared with a special type of Kolmogorov model. In the case of quadratic harvesting, the fixed points are computed after nondimensionalization. For the non-existence of periodic orbits in the first quadrant we apply a condition of the general Kolmogorov model to exist a Dulac function. We show that this system does not have periodic orbits with the help of numerical simulation and graphical representation.

Study on State Feedback Impulsive Control of Red Squirrel Protection with the Influence of Herpes Virus

A kind of state feedback nonlinear impulse model was established to describe the inter-specific competition between red squirrel and grey squirrel and the artificial protection of red squirrel.The effects of releasing artificial red squirrel individuals and eliminating grey individuals on the persistence of red squirrel population were studied. The existence and stability of the order-1 periodic solution of the impulsive control model was proved by using the method of successor function and Dulac function. The numerical simulation is given to verify the theoretical results.

Global dynamics in a tumor-immune model with an immune checkpoint inhibitor

In this paper, we fill several key gaps in the study of the global dynamics of a highly nonlinear tumor-immune model with an immune checkpoint inhibitor proposed by Nikolopoulou et al. (Letters in Biomathematics, 5 (2018), S137-S159). For this tumour-immune interaction model, it is known that the model has a unique tumour-free equilibrium and at most two tumorous equilibria. We present sufficient and necessary conditions for the global stability of the tumour-free equilibrium or the unique tumorous equilibrium. The global dynamics is obtained by employing a new Dulac function to establish the nonexistence of nontrivial positive periodic orbits. Our analysis shows that we can almost completely classify the global dynamics of the model with two critical values \begin{document}$ C_{K0}, C_{K1} (C_{K0}>C_{K1}) $\end{document} for the carrying capacity \begin{document}$ C_K $\end{document} of tumour cells and one critical value \begin{document}$ d_{T0} $\end{document} for the death rate \begin{document}$ d_{T} $\end{document} of T cells. Specifically, the following are true. (ⅰ) When no tumorous equilibrium exists, the tumour-free equilibrium is globally asymptotically stable. (ⅱ) When \begin{document}$ C_K \leq C_{K1} $\end{document} and \begin{document}$ d_T>d_{T0} $\end{document} , the unique tumorous equilibrium is globally asymptotically stable. (ⅲ) When \begin{document}$ C_K >C_{K1} $\end{document} , the model exhibits saddle-node bifurcation of tumorous equilibria. In this case, we show that when a unique tumorous equilibrium exists, tumor cells can persist for all positive initial densities, or can be eliminated for some initial densities and persist for other initial densities. When two distinct tumorous equilibria exist, we show that the model exhibits bistable phenomenon, and tumor cells have alternative fates depending on the positive initial densities. (ⅳ) When \begin{document}$ C_K > C_{K0} $\end{document} and \begin{document}$ d_T = d_{T0} $\end{document} , or \begin{document}$ d_T>d_{T0} $\end{document} , tumor cells will persist for all positive initial densities.

Elemental access to limit cycle existence in Biomath education.

This paper originated from the desire to develop elementary calculus based tools to empower students, not necessarily with a strong mathematical background, to test predator-prey related models for boundedness of solutions and for the existence of limit cycles. There are several well-known methods available to prove, or disprove, the existence of bounded solutions to systems of differential equations. These methods rely on LiГ©nard's theorem or using Dulac or Lyaponov functions. The level of mathematics required in the study of differential equations is not addressed in the courses presented on the first year level, and students in biology, ecology, economics and other fields are often not suitably equipped to perform these advanced techniques.The conditions under which a unique limit cycle exists in predator-prey systems is considered a primary problem in mathematical ecology. A great deal of mathematical effort has gone into trying to establish simple, yet general, theorems which will allow one to decide whether a given set of nonlinear equations has a limit cycle or not. We introduce a method to first determine the boundedness of solution trajectories in such a way that the transformation to a LiГ©nard system or the use of a Dulac function can be avoided. Once boundedness of trajectories has been established, the nature of the equilibrium points reduces to simple eigenvalue analysis. The Elemental Limit Cycle method (ELC) provides elementary criteria to evaluate the nature of the pivotal functions of a system which will indicate boundedness and may be applicable to more general models.

MODELLING THE DYNAMICS OF AVIAN INFLUENZA WITH NONLINEAR RECOVERY RATE AND PSYCHOLOGICAL EFFECT

In this paper, a SI-SEIR type avian influenza epidemic model with psychological effect, nonlinear recovery rate and saturation inhibition effect is formulated to study the transmission and control of avian influenza virus. By setting the basic reproductive number as the threshold parameter and constructing Lyapunov function, Dulac function and using the Li-Muldowney's geometry approach, we prove the local and global stability of disease-free equilibria and endemic equilibrium. Theoretical analysis are carried out to show the role of the saturation inhibition effect, psychological effect and effective medical resources in this model, and numerical simulations are also given to verify the results.

Mathematical study of SIR epidemic model under convex incidence rate

In this manuscript, we examine the SIR model under convex incidence rate. We first formulate the famous SIR model under the aforesaid incidence rate. Further, we develop some sufficient analysis to examine the dynamical behavior of the model under consideration. We compute the basic reproductive number $\mathcal{R}_0.$ Also we study the global attractivity results via using Dulac function theory. Further, we also provide some information about the stability of the endemic and disease free equilibria for the considered model. In addition, we use nonstandard finite difference scheme to perform numerical simulation of the considered model via using Matlab. We provide different numerical plots for two different values of contact rate and taking various initial values for compartments involved in the considered model.

Global stability in a modified Leslie-Gower type predation model assuming mutual interference among generalist predators.

In the ecological literature, mutual interference (self-interference) or competition among predators (CAP) to effect the harvesting of their prey has been modeled through different mathematical formulations. In this work, the dynamical properties of a Leslie-Gower type predation model is analyzed, incorporating one of these forms, which is described by the function $g\left(y\right) =y^{\beta }$, with $0<\beta <1$. This function $g$ is not differentiable for $y=0$, and neither the Jacobian matrix of the system is not defined in the equilibrium points over the horizontal axis ($x-axis$). To determine the nature of these points, we had to use a non-standard methodology. Previously, we have shown the fundamental properties of the Leslie-Gower type model with generalist predators, to carry out an adequate comparative analysis with the model where the competition among predators (CAP) is incorporated. The main obtained outcomes in both systems are: (i) The unique positive equilibrium point, when exists, is globally asymptotically stable (g.a.s), which is proven using a suitable Lyapunov function. (ii) There not exist periodic orbits, which was proved constructing an adequate Dulac function.

Global dynamics and sliding motion in A(H7N9) epidemic models with limited resources and Filippov control

This paper investigates the effects of different control measures on the dynamics of A(H7N9) virus transmission from poultry to human. An SI-SIR model with nonlinear incidence and recovery rates is formulated to evaluate the combined impacts of government intervention strategies and available hospital resources. The global stability of the disease-free and endemic equilibria is obtained by constructing Lyapunov and Dulac functions. Furthermore, based on the threshold policy, the SI-SIR model is extended to a novel three-dimensional Filippov one to represent the control measures being triggered once the total number of infected poultry and human reaches the tolerant level Ic. Model solutions are able to approach either one real equilibrium or the pseudo-equilibrium, depending on the tolerant threshold. Our results suggest that in order to diminish the outbreak of A(H7N9) virus or lead the number of infections to an expected level, it requires not only adequate hospital resources and certain government interventions, but also a good threshold policy.

Fixed and moving limit cycles for Liénard equations

We consider a family of planar vector fields that writes as a Lienard system in suitable coordinates. It has a fixed closed invariant curve that often contains periodic orbits of the system. We prove a general result that gives the hyperbolicity of these periodic orbits, and we also study the coexistence of them with other periodic orbits. Our family contains the celebrated Wilson polynomial Lienard equation, as well as all polynomial Lienard systems having hyperelliptic limit cycles. As an illustrative example, we study in more detail a natural 1-parametric extension of Wilson example. It has at least two limit cycles, one of them fixed and algebraic and the other one moving with the parameter, presents a transcritical bifurcation of limit cycles and for a given parameter has a non-hyperbolic double algebraic limit cycle. In order to prove that for some values of the parameter the system has exactly two hyperbolic limit cycles, we use several suitable Dulac functions.

A note on the existence and construction of Dulac functions

This paper is concerned with a predator–prey model with hyperbolic mortality and prey harvesting. The parameter regions for the stability and instability of the unique positive constant solution of ODE and PDE are derived, respectively. Especially, the global asymptotical stability of positive constant equilibrium of the diffusive model is obtained by iterative technique. The stability and direction of periodic solutions of ODE and PDE are investigated by center manifold theorem and normal form theory, respectively. Numerical simulations are carried out to depict our theoretical analysis.