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## The dynamical behaviors of a nonautonomous Holling III predator-prey system with impulses

### Journal of Applied Mathematics and Computing (2015-02-01) 47: 193-209 , February 01, 2015

In this paper, we study a Holling III nonautonomous predator-prey system with impulses. Firstly, the dynamics of single-species nonautonomous Logistic system with impulses are discussed. Further, based on these results and by impulsive differential comparison theorem, the extinction and permanence of Holling III nonautonomous predator-prey system with impulses are obtained.

## Positive periodic solutions of discrete Lotka–Volterra cooperative systems with delays

### Acta Mathematica Vietnamica (2013-09-01) 38: 461-470 , September 01, 2013

The existence of positive periodic solutions of discrete nonautonomous Lotka–Volterra cooperative systems with delays is studied by applying the continuation theorem of coincidence degree theory.

## Dynamic formation of oriented patches in chondrocyte cell cultures

### Journal of Mathematical Biology (2011-10-01) 63: 757-777 , October 01, 2011

Growth factors have a significant impact not only on the growth dynamics but also on the phenotype of chondrocytes (Barbero et al. in J. Cell. Phys. 204:830–838, 2005). In particular, as chondrocytes approach confluence, the cells tend to align and form coherent patches. Starting from a mathematical model for fibroblast populations at equilibrium (Mogilner et al. in Physica D 89:346–367, 1996), a dynamic continuum model with logistic growth is developed. Both linear stability analysis and numerical solutions of the time-dependent nonlinear integro-partial differential equation are used to identify the key parameters that lead to pattern formation in the model. The numerical results are compared quantitatively to experimental data by extracting statistical information on orientation, density and patch size through Gabor filters.

## Bifurcation Analysis of a Predator–Prey System with Generalised Holling Type III Functional Response

### Journal of Dynamics and Differential Equations (2008-09-01) 20: 535-571 , September 01, 2008

We consider a generalised Gause predator–prey system with a generalised Holling response function of type III:
$$p(x) = \frac{mx^2}{ax^2+bx+1}$$
. We study the cases where *b* is positive or negative. We make a complete study of the bifurcation of the singular points including: the Hopf bifurcation of codimensions 1 and 2, the Bogdanov–Takens bifurcation of codimensions 2 and 3. Numerical simulations are given to calculate the homoclinic orbit of the system. Based on the results obtained, a bifurcation diagram is conjectured and a biological interpretation is given.

## Solvability of impulsive $$(n,n-p)$$ ( n , n - p ) boundary value problems for higher order fractional differential equations

### Mathematical Sciences (2016-09-01) 10: 71-81 , September 01, 2016

We present a new general method for converting an impulsive fractional differential equation to an equivalent integral equation. Using this method and employing a fixed point theorem in Banach space, we establish existence results of solutions for a boundary value problem of impulsive singular higher order fractional differential equation. An example is presented to illustrate the efficiency of the results obtained. A conclusion section is given at the end of the paper.

## Regularity of Transition Fronts in Nonlocal Dispersal Evolution Equations

### Journal of Dynamics and Differential Equations (2016-03-11): 1-32 , March 11, 2016

It is known that solutions of nonlocal dispersal evolution equations do not become smoother in space as time elapses. This lack of space regularity would cause a lot of difficulties in studying transition fronts in nonlocal equations. In the present paper, we establish some general criteria concerning space regularity of transition fronts in nonlocal dispersal evolution equations with a large class of nonlinearities, which allows the applicability of various techniques for reaction–diffusion equations to nonlocal equations, and hence serves as an initial and fundamental step for further studying various important qualitative properties of transition fronts such as stability, uniqueness and asymptotic speeds. We also prove the existence of continuously differentiable and increasing interface location functions, which give a better characterization of the propagation of transition fronts and are of great technical importance.

## Finite element approximation of spatially extended predator–prey interactions with the Holling type II functional response

### Numerische Mathematik (2007-10-01) 107: 641-667 , October 01, 2007

We study the numerical approximation of the solutions of a class of nonlinear reaction–diffusion systems modelling predator–prey interactions, where the local growth of prey is logistic and the predator displays the Holling type II functional response. The fully discrete scheme results from a finite element discretisation in space (with lumped mass) and a semi-implicit discretisation in time. We establish a priori estimates and error bounds for the semi discrete and fully discrete finite element approximations. Numerical results illustrating the theoretical results and spatiotemporal phenomena are presented in one and two space dimensions. The class of problems studied in this paper are real experimental systems where the parameters are associated with real kinetics, expressed in nondimensional form. The theoretical techniques were adapted from a previous study of an idealised reaction–diffusion system (Garvie and Blowey in Eur J Appl Math 16(5):621–646, 2005).

## Dynamics of HIV infection in lymphoid tissue network

### Journal of Mathematical Biology (2016-03-01) 72: 909-938 , March 01, 2016

Human immunodeficiency virus (HIV) is a fast replicating ribonucleic acid virus, which can easily mutate in order to escape the effects of drug administration. Hence, understanding the basic mechanisms underlying HIV persistence in the body is essential in the development of new therapies that could eradicate HIV infection. Lymphoid tissues are the primary sites of HIV infection. Despite the recent progress in real-time monitoring technology, HIV infection dynamics in a whole body is unknown. Mathematical modeling and simulations provide speculations on global behavior of HIV infection in the lymphatic system. We propose a new mathematical model that describes the spread of HIV infection throughout the lymphoid tissue network. In order to represent the volume difference between lymphoid tissues, we propose the proportionality of several kinetic parameters to the lymphoid tissues’ volume distribution. Under this assumption, we perform extensive numerical computations in order to simulate the spread of HIV infection in the lymphoid tissue network. Numerical computations simulate single drug treatments of an HIV infection. One of the important biological speculations derived from this study is a drug saturation effect generated by lymphoid network connection. This implies that a portion of reservoir lymphoid tissues to which drug is not sufficiently delivered would inhibit HIV eradication despite of extensive drug injection.

## Bistability induced by generalist natural enemies can reverse pest invasions

### Journal of Mathematical Biology (2017-01-17): 1-33 , January 17, 2017

Analytical modeling of predator–prey systems has shown that specialist natural enemies can slow, stop and even reverse pest invasions, assuming that the prey population displays a strong Allee effect in its growth. We aimed to formalize the conditions in which spatial biological control can be achieved by generalists, through an analytical approach based on reaction–diffusion equations. Using comparison principles, we obtain sufficient conditions for control and for invasion, based on scalar bistable partial differential equations. The ability of generalist predators to control prey populations with logistic growth lies in the bistable dynamics of the coupled system, rather than in the bistability of prey-only dynamics as observed for specialist predators attacking prey populations displaying Allee effects. As a consequence, prey control is predicted to be possible when space is considered in additional situations other than those identified without considering space. The reverse situation is also possible. None of these considerations apply to spatial predator–prey systems with specialist natural enemies.

## Extinction of a two species competitive system with nonlinear inter-inhibition terms and one toxin producing phytoplankton

### Advances in Difference Equations (2016-10-12) 2016: 1-13 , October 12, 2016

A two species non-autonomous competitive phytoplankton system with nonlinear inter-inhibition terms and one toxin producing phytoplankton is studied in this paper. Sufficient conditions which guarantee the extinction of a species and the global attractivity of the other one are obtained. Some parallel results corresponding to Yue (Adv. Differ. Equ. 2016:1, 2016, doi: 10.1007/s11590-013-0708-4 ) are established. Numeric simulations are carried out to show the feasibility of our results.