# Limit theorems for multi-type general branching processes with population dependence

## TL;DR

This paper develops limit theorems for multi-type population models influenced by population size and composition, providing deterministic and stochastic approximations, and extends the framework to sexual reproduction systems.

## Contribution

It introduces a comprehensive measure-valued process framework for multi-type populations with population dependence, deriving Law of Large Numbers and Central Limit Theorem results.

## Key findings

- Established a deterministic approximation via Law of Large Numbers.
- Derived a Central Limit Theorem for fluctuations around the deterministic limit.
- Extended the model to include sexual reproduction and monogamous mating systems.

## Abstract

A general multi-type population model is considered, where individuals live and reproduce according to their age and type, but also under the influence of the size and composition of the entire population. We describe the dynamics of the population density as a measure-valued process and obtain its asymptotics, as the population grows with the environmental carrying capacity. "Density" in this paper generally refers to the population size as compared to the carrying capacity. Thus, a deterministic approximation is given, in the form of a Law of Large Numbers, as well as a Central Limit Theorem. Migration can also be incorporated. This general framework is then adapted to model sexual reproduction, with a special section on serial monogamic mating systems.