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Encyclopedia of Quality of Life and Well-Being Research

This book is devoted to the study of maximum principles in partial differential equations. I t contains a wealth of material much of which is presented for the first time in a book form. An attractive feature of the book is that it is completely elementary and thus accessible to a wide audience of readers. The book has four chapters. Chapter I deals with the one dimensional maximum principle. The discussion of this very simple model of a maximum principle forms a good introduction to the general theory. Various applications of the principle are given to show that it is a very useful tool even in the study of ordinary differential equations. As an example, it is shown that many oscillation and comparison results in the Sturm-Liouville theory could be deduced most easily by a maximum principle argument. The proper discussion of maximum principles in partial differential equations begins in Chapter II . This chapter, which is the backbone of the book, is devoted to elliptic equations. The material covered in this chapter includes the E. Hopf maximum principle and its generalizations; the Phragmèn-Lindelöf principle for solutions of elliptic equations; Serrin's version of the Harnack inequality for solutions of general elliptic equations in two variables (this is probably the most difficult result discussed in the book) ; various versions of the Hadamard three circles theorems for solutions of elliptic equations; applications of the maximum principle to nonlinear equations and to problems of fluid flow. Chapter III is devoted to parabolic equations. The plan of this chapter parallels that of Chapter II . The topics discussed include the L. Nirenberg strong maximum principle; a three curves theorem with an interesting application to the Tikhonov uniqueness theorem; a Phragmèn-Lindelöf principle for parabolic equations with applications to uniqueness results; nonlinear operators; a maximum principle for certain parabolic systems. The fourth and the last chapter is devoted to hyperbolic equations. The results in this chapter are somewhat special since a maximum principle in the proper sense does not hold for solutions of hyperbolic equations. Nevertheless, solutions of certain hyperbolic equations

Maximum Principles In Differential Equations

Finite Element Methods And Their Applications

Monotone dynamical systems Stability and convergence Competitive and cooperative differential equations Irreducible cooperative systems Cooperative systems of delay differential equations Nonquasimonotone delay differential equations Quasimonotone systems of parabolic equations A competition model Appendix Bibliography.

Monotone Dynamical Systems: An Introduction To The Theory Of Competitive And Cooperative Systems (Mathematical Surveys And Monographs) By Hal L. Smith

Persistence and extinction in spatial models with a carrying capacity driven diffusion and harvesting

In the last two decades the world had faced three respiratory syndrome outbreaks incurred by Coronavirus. Though the wild animals are the primary carriers of the virus, the human population managed to survive sacrificing more than 1, 600 lives from 2002 to 2012. But the current virus outbreak has already taken more than 2, 462 lives since 22 February 2020. In the first few days, when the cases were being introduced under light, there were no treatment for the infection and the unleashed spread demands to be analyzed to see the pattern of the outbreak. This manuscript aims to look into the growth map of the COVID-19 outbreak under mathematical growth functions and tries to understand which growth pattern assembles the scenario for the cases.

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Coronavirus Outbreak and the Mathematical Growth Map of COVID-19

We study the interaction between different types of dispersal, intrinsic growth rates and carrying capacities of two competing species in a heterogeneous environment: one of them is subject to a regular diffusion while the other moves in the direction of most per capita available resources. If spatially heterogeneous carrying capacities coincide, and intrinsic growth rates are proportional then competitive exclusion of a regularly diffusing population is inevitable. However, the situation may change if intrinsic growth rates for the two populations have different spatial forms. We also consider the case when carrying capacities are different. If the carrying capacity of a regularly diffusing population is higher than for the other species, the two populations may coexist; as the difference between the two carrying capacities grows, competitive exclusion of the species with a lower carrying capacity occurs.

Competitive spatially distributed population dynamics models: Does diversity in diffusion strategies promote coexistence?

In the promptness of the COVID-19 outbreak, it would be very important to observe and estimate the pattern of diseases to reduce the contagious infection. To study this effect, we developed a COVID-19 epidemic model that incorporates five various groups of individuals. Then we analyze the model by evaluating the equilibrium points and analyzing their stability as well as determining the basic reproduction number. Also, numerical simulations show the dynamics of a different group of the population over time. Thus, our findings based on the sensitivity analysis and the reproduction number highlight the role of outbreak of the virus that can be useful to avoid a massive collapse in Bangladesh and rest of the world. The outcome of this study concludes that outbreak will be in control which ensure the social and economic stability.

/pdf/covid-19-epidemic-compartments-model-and-bangladesh-1ab2g63iw6.pdf

COVID-19 Epidemic Compartments Model and Bangladesh

We study a Lotka system describing two competing populations, and each of them chooses its diffusion strategy as the tendency to have a distribution proportional to a certain positive prescribed function. For instance, the standard diffusion corresponds to the choice of a uniform distribution. The paper is focused on the interplay of species competition and diffusion strategies. In the case when one of the diffusion strategies is proportional to the carrying capacity, while the other is not, and the competition does not discriminate the former species, we prove the competitive exclusion of the latter one. If the competition favors the latter species, there is still a range of parameters for which there is a coexistence, thanks to the better dispersal strategy chosen by the former species. The dependency on the interaction type, diffusion coefficients and intrinsic growth rates is explored. We prove that in the limit case, higher diffusion coefficients are detrimental while higher growth rates, as well as lower resources sharing, are beneficial for population survival.

Md. Kamrujjaman

Papers

Persistence and extinction in spatial models with a carrying capacity driven diffusion and harvesting

Coronavirus Outbreak and the Mathematical Growth Map of COVID-19

Competitive spatially distributed population dynamics models: Does diversity in diffusion strategies promote coexistence?

COVID-19 Epidemic Compartments Model and Bangladesh

Lotka systems with directed dispersal dynamics: Competition and influence of diffusion strategies