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By
Yan, CongNi; Dong, LingZhen; Liu, Ming
3 Citations
In this paper, we study a Holling III nonautonomous predatorprey system with impulses. Firstly, the dynamics of singlespecies 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 predatorprey system with impulses are obtained.
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By
Anh, Trinh Tuan
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.
By
Grote, Marcus J.; Palumberi, Viviana; Wagner, Barbara; Barbero, Andrea; Martin, Ivan
Show all (5)
3 Citations
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 timedependent nonlinear integropartial 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.
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By
Lamontagne, Yann; Coutu, Caroline; Rousseau, Christiane
33 Citations
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.
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By
Liu, Yuji
2 Citations
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.
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By
Speed, Maria Simonsen ; Balding, David Joseph ; Hobolth, Asger
In population genetics, the Dirichlet (also called the Balding–Nichols) model has for 20 years been considered the key model to approximate the distribution of allele fractions within populations in a multiallelic setting. It has often been noted that the Dirichlet assumption is approximate because positive correlations among alleles cannot be accommodated under the Dirichlet model. However, the validity of the Dirichlet distribution has never been systematically investigated in a general framework. This paper attempts to address this problem by providing a general overview of how allele fraction data under the most common multiallelic mutational structures should be modeled. The Dirichlet and alternative models are investigated by simulating allele fractions from a diffusion approximation of the multiallelic Wright–Fisher process with mutation, and applying a momentbased analysis method. The study shows that the optimal modeling strategy for the distribution of allele fractions depends on the specific mutation process. The Dirichlet model is only an exceptionally good approximation for the pure drift, Jukes–Cantor and parentindependent mutation processes with small mutation rates. Alternative models are required and proposed for the other mutation processes, such as a Beta–Dirichlet model for the infinite alleles mutation process, and a Hierarchical Beta model for the Kimura, Hasegawa–Kishino–Yano and Tamura–Nei processes. Finally, a novel Hierarchical Beta approximation is developed, a Pyramidal Hierarchical Beta model, for the generalized timereversible and singlestep mutation processes.
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By
Garvie, Marcus R.; Trenchea, Catalin
24 Citations
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 semiimplicit 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).
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By
Kong, Jude D.; Salceanu, Paul; Wang, Hao
1 Citations
Biodegradation, the disintegration of organic matter by microorganism, is essential for the cycling of environmental organic matter. Understanding and predicting the dynamics of this biodegradation have increasingly gained attention from the industries and government regulators. Since changes in environmental organic matter are strenuous to measure, mathematical models are essential in understanding and predicting the dynamics of organic matters. Empirical evidence suggests that grazers’ preying activity on microorganism helps to facilitate biodegradation. In this paper, we formulate and investigate a stoichiometrybased organic matter decomposition model in a chemostat culture that incorporates the dynamics of grazers. We determine the criteria for the uniform persistence and extinction of the species and chemicals. Our results show that (1) if at the unique internal steady state, the per capita growth rate of bacteria is greater than the sum of the bacteria’s death and dilution rates, then the bacteria will persist uniformly; (2) if in addition to this, (a) the grazers’ per capita growth rate is greater than the sum of the dilution rate and grazers’ death rate, and (b) the death rate of bacteria is less than some threshold, then the grazers will persist uniformly. These conditions can be achieved simultaneously if there are sufficient resources in the feed bottle. As opposed to the microcosm decomposition models’ results, in a chemostat culture, chemicals always persist. Besides the transcritical bifurcation observed in microcosm models, our chemostat model exhibits Hopf bifurcation and Rosenzweig’s paradox of enrichment phenomenon. Our sensitivity analysis suggests that the most effective way to facilitate degradation is to decrease the dilution rate.
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By
Labbas, Rabah; Medeghri, Ahmed ; Menad, Abdallah
In this work, we study an elliptic differential equation set in three habitats with skewness boundary conditions at the interfaces. It represents the linear stationary case of dispersal problems of population dynamics which incorporate responses at interfaces between the habitats. Existence, uniqueness and regularity of the solution of these problems are obtained in Hölder spaces under necessary and sufficient conditions on the data. Our techniques are based on the semigroup theory, the fractional powers of linear operators, the
$$H^{\infty }$$
functional calculus for sectorial operators in Banach spaces and some properties of real interpolation spaces.
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By
Guo, Shangjiang ; Zimmer, Johannes
This paper deals with the existence, monotonicity, uniqueness, and asymptotic behaviour of travelling wavefronts for a class of temporally delayed, spatially nonlocal diffusion equations.
