Mathematical modelling-Malthus population problem

The problem of catalogue the logical regression model of Malthus population model

1. The question was raised

Malthus's demographic problem was a prediction of group growth, presented by Malthus, and he wrote a book on population growth, and the whole book was studied in the same way as Euclid studied geometry, and the two basic axioms he proposed were: food is necessary for human survival, and will maintain the status quo.

Thus, we can establish the corresponding research object: T→p (t) t \rightarrow P (t) where T is the time, p (t) p (t) represents the population of the moment T, we think that the population, although the time changes, of course, it certainly is affected by other factors, this is only our simplification of this problem.

There are two goals we want to achieve: predict the number of people in the future at a certain T-moment. When T→∞t \rightarrow \infty, how many people are there.
This is our problem, how to solve it.

2. The Malthus population model

Gowers also discussed the problem in mathematics, where the population can be expressed as a number pair: (T,p (t)) (T,p (t)) where T stands for the moment, p (t) p (t) represents the population of the moment T. We also use B,d to denote birth and death rates.
If the total population of 2002 years is p, the number of births and deaths that 2002 years is BP,DP, so the total number of 2003 years is: p+bp−dp= (1+b−d) p= (1+r) p p + BP-DP = (1 + b-d) p = (1 + r) p Calculation, is a discrete model, because the span of time is one year, but the birth and death of the population at different times have, in order to turn it into a continuous model, we use the idea of integral to calculate a short period of time, the population growth situation, can have: P (t+∇t) −p (t) =rp (t) ∇t p (t + \nabla T)-P (t) = RP (t) \nabla T in ∇t \nabla t This period of population growth is: growth rate    x   t time of population   &nb Sp x    time increase \ \ \ x\ \ t time population \ \ \ x\ \ \ Time put ∇t \nabla t except for: P (t+∇t) −p (t) ∇t=rp (t) \frac{p (t + \nabla t)- P (t)}{\nabla T} = R P (t) takes the limit of ∇t→0 \nabla T \rightarrow 0 to obtain the differential equation: DP (t) dt=rp (t)