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Published in:
2014 | OriginalPaper | Chapter
5. Food-Limited Population Models
Authors : Ravi P. Agarwal, Donal O’Regan, Samir H. Saker
Published in: Oscillation and Stability of Delay Models in Biology
Publisher: Springer International Publishing
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Abstract
Smith [66] reasoned that a food-limited population in its growing stage requires food for both maintenance and growth, whereas, when the population has reached saturation level, food is needed for maintenance only. On the basis of these assumptions, Smith derived a model of the form which is called the “food limited” population. Here N, r, and K are the mass of the population, the rate of increase with unlimited food, and the value of N at saturation, respectively. The constant 1∕c is the rate of replacement of mass in the population at saturation. Since a realistic model must include some of the past history of the population, Gopalsamy, Kulenovic and Ladas introduced the delay in (5.1) and considered the equation as the delay “food-limited” population model, where r, K, c, and τ are positive constants.
$$\displaystyle{ \frac{dN(t)} {dt} = rN(t) \frac{K - N(t)} {K + crN(t)} }$$
(5.1)
$$\displaystyle{ \frac{dN(t)} {dt} = rN(t) \frac{K - N(t-\tau )} {K + crN(t-\tau )}, }$$