Struggle for Existence?

A struggle for existence inevitably follows from the high rate at which all organic beings tend to increase. Every being, which during its natural lifetime produces several eggs or seeds, must suffer destruction during some period of its life, and during some season or occasional year; otherwise, on the principle of geometrical increase, its number would quickly become so inordinately great that no country could support the product. Hence, as more individuals are produced than can possibly survive, there must in every case be a struggle for existence, either one individual with another of the same species, or with the individuals of distinct species, or with physical conditions of life. It is the doctrine of Malthus applied with manifold force to the whole animal and vegetable kingdoms; for in this case there can be no artificial increase of food, and no prudential restraint from marriage. Although some species may be now increasing, more or less rapidly, in numbers, all cannot do so, for the world would not hold them…. 
The amount of food for each species of course gives the extreme limit to which each can increase: but very frequently it is not the obtaining food, but the serving as prey to other animals, which determines the average numbers of a species.
 Charles Darwin 
以上是摘自Darwin一段對族群數目的敘述，Darwin雖然不是數學家，但他在生物模式的深刻洞見，使他了解族群數目問題本質。依照老師課堂上對數學模型的分析，試著寫出能描述Darwin這段文字的方程式，並分析這些方程式模式的內涵。(hint: doctrine of Malthus, logistic equation, and other models)。請引用上述文字的片段於整理的數學模式。