Abstract
In this paper we explore the possibility of using the scientific computing method to obtain the inverse B-Transform of Oyelami and Ale [1]. Using some suitable conditions and the symbolic programming method in Maple 15 we obtained the asymptotic expansion for the inverse B-transform then used the residue theorem to obtain solutions of Impulsive Diffusion and Von-Foerster-Makendrick models. The results obtained suggest that drugs that are needed for prophylactic or chemotherapeutic purposing the concentration must not be allowed to oscillate about the steady state. Drugs that are to be used for immunization should not oscillate at steady state in order to have long residue effect in the blood. From Von-Foerster-Makendrick model, we obtained the conditions for population of the specie to attain super saturation level through the “dying effect” phenomenon ([2-4]). We used this phenomenon to establish that the environment cannot accommodate the population of the specie anymore which mean that a catastrophic stage t * is reached that only the fittest can survive beyond this regime (i.e. t > t * ) and that there would be sharp competition for food, shelter and
Highlights
Impulsive differential equations (IDEs) describe processes which quickly change their states for short moment of times when compare to the total evolution time for the systems
The B-transform was developed to take into consideration the impulsive effect of the system to provide a method for finding solutions for the fixed moments IDEs ([1,11,12,13])
More works need to be done on applications of B-transform especially taxism problems by studying the response of living organisms to stimulus
Summary
Impulsive differential equations (IDEs) describe processes which quickly change their states for short moment of times when compare to the total evolution time for the systems. The impulses may happen at fixed or non-fixed moments and the behavior of state variables describing the processes may show some “jumps”, “shocks” attributes etc This kind of impulsive behavior makes the IDEs not accessible to most existing concepts and theories in differential equations, ecology, biomathematics, engineering and control systems ([2,3,4,5,6,7,8,9]). We must emphasize that under certain conditions like rapid changes i.e. advent of war, earthquake, displacement of persons etc., the population tends to be impulsive in nature (see [2,8,16,17,18]) For this reason, we consider impulsive analog of the Von-Foerster-Makendrick model of an age-dependent population in given ecosystem. The model is typical impulsive partial differential equations and has potential applications in modelling the population of species stratified into age groups and epidemiology of infectious diseases like malaria and HIV/AIDs
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