Question

Consider a population that grows according to the Beverton Holt updating function and is harvested at a linear rate h ≥ 0 The number of individuals per square kilometer of the species satisfies the DTDSxₜ₊₁ =13xₜ/1+xₜ - hxₜfor t = 0,1,2,....

Give the updating function for this DTDS. Your answer will include the parameter ℎ
Answer: f(x) =____

Answer 1

The updating function for a population modeled by a Beverton-Holt function with linear harvesting is f(x) = (13x) / (1 + x) - hx. This incorporates both the growth and harvest factors affecting the population size.

Step-by-step explanation:

The question involves deriving the updating function for a population model that incorporates both growth and harvesting influences. The Beverton-Holt updating function, which is a type of discrete-time difference equation, is used to model the population. Harvesting, at a linear rate h, affects the population by reducing it in proportion to its size. The updating function in question is described as DTDSxt+1 = 13xt/(1+xt) - hxt for t = 0,1,2, and so on.

To write the updating function, we simply need to express this equation in terms of a function f(x). The formula provided in the question already essentially gives us the updating function, except it presents it as a sequence rather than explicitly in function format. The updating function for the population based on the given equation will be:

f(x) = (13x) / (1 + x) - hx

This function captures the dynamics of population growth balanced by a linear harvest rate. The growth factor (13x) / (1 + x), represents the growth according to the Beverton-Holt model, while the harvesting term, hx, directly subtracts from the population based on the current population size and the harvesting rate h.