Introduction to Mathematical Biology

Abstract

In Chapter 2 we considered the chemostat model and used mathematics to answer the question: How should we choose the outflow rate in order to harvest the maximum amount of bacteria. Our model however was incomplete because we assumed that the nutrient concentration in the growth chamber is constant in time, and hence our answer is questionable. In the present chapter we want to correct the answer, by basing it on a more complete mathematical model of the chemostat.We begin by introducing the following notation: V = volume of the bacterial chamber, C(t) = concentration of nutrients in the chamber, C 0 = constant concentration of nutrients supply, r = rate of inflow and outflow, B(t) = concentration of the bacteria in the chamber.We assume that mass of the bacteria formed mass of the nutrients used = constant = γ; γ is the yield constant; γ < 1 since bacteria of mass 1 is formed by consumption of nutrients of larger mass 1/γ. By conservation of nutrient mass, rate of change = input − washout − consumption.Based on experimental evidence we take the rate of bacterial growth in the entire bacterial chamber to be m 0 C a +C B,Electronic supplementary material The online version of this chapter

Numerical Analysis of a Fractional Order Discrete Prey – Predator System with Functional Response

“…Classical basic Lotka -Volterra model [4] have been improved by many authors by introducing functional responses and harvesting. In the two species interactions, Holling type II functional response expressed by () ax fx bx   [3,5] plays a vital role.…”

“…Models in population dynamics are studied with ordinary differential equations, fractional order differential equations, difference equations and diffusion equations. The fractional order differential equations has wide applications in mathematical biology [7,8] and other interdisciplinary fields [4] due to their ability to accommodate memory effect.…”

Free boundary problems arising in biology

“…The final step is to verify that these predictions agree with known experimental results and, at the same time, can be used to address the biological question. Such an approach was adopted in two recent textbooks in mathematical biology, one for undergraduates, using ODEs [1], and another for graduate students using also PDEs [9].…”

Fractional‐Order Model of Two‐Prey One‐Predator System

“…is the equilibrium point of system (4). Solving the equations of system (12), we get the following equilibrium points.…”

“…Mathematical biology tries to model, study, analyze, and interpret biological phenomenon such as the interactions, coexistence, and evolution of different species [1][2][3][4]. These interactions may be among the individuals of the same species and among the individuals of different species or interactions against the environment, disease, and food supply.…”