Population dynamics: formulating a teory

From the documentary "Galapagos - Die Invasion der Aliens" by Felix Heidinger and A. Udo Zimmermann: The Galapagos Islands had 300 million years to allow the few settlers, plants and animals, which had naturally come to the islands, to develop in a unique living space. Only a few came to these remote islands, and under the rough conditions on this volcano archipelago with almost no competition, a few natural hunters could slowly develop. Approximately 400 years ago, this balance was changed. Men visited the islands and brought their domestic animals with them, such as cattle, goats, pigs, dogs and cats. Other animals, such as rats or mice arrived as blind passengers on the sailing ships. Some of these imported animals escaped their owners and formed a wild population. For the fauna of the Galapagos, these aliens, these “foreigners from another world”, present a great danger. For example, cottony cushion scales, pests of citrus were imported accidentally with a load of rotten lemons and entered immediately into competition with other insects, altering the ecosystem. The solution found by the authorities to restore the initial situation has been to introduce another alien species: the ladybug, that feeds on scale insects. The idea was that, once all the scale insects had been killed , ladybugs would be eliminated due to lack of food: two birds with one stone. But things will really happen that way? To describe several competing species is extremely difficult, so we will deal with the simplest task to formulate a theory that explains the evolution in time of the number of individuals (the numerosity) in a isolated population (i.e. considered as if it were the only existing). We will use the scientific method: the theory will be formulated on the basis of observations or plausible assumptions, then his predictions will be subjected to criticism to correct any controversial aspects. Let us start by formulating an initial theory.