## MATH0030｜Mathematical Ecology数学生态学 伦敦大学学院

statistics-labTM为您提供伦敦大学学院（University College London，简称：UCL）Mathematical Ecology数学生态学英国代写代考辅导服务！

Mathematical models are used extensively in many areas of the Biological Sciences. This course
aims to give a sample of the construction and mathematical analysis of such models in Population Ecology. The fundamental question to be addressed is: what natural (or human) factors
control the abundance and distribution of the various populations of animals and plants that
we see in Nature?
No special knowledge of Ecology is required or assumed. However, an interest in, and willingness
to learn about, concepts and problems in this area are essential. Mathematical techniques used
include calculus, mathematical methods and linear algebra, and those developed include the
important qualitative technique of phase plane analysis which the course uses extensively.

## Mathematical Ecology数学生态学案例

The general deterministic model for the dynamics of a single, unstructured population reads:
$$\dot{N}(t)=f(N, t)=f(N(t), t)$$
$N(t) \in \mathbf{R}$ is usually interpreted as population density rather than the total size.

The most basic model is exponential growth with
$$\dot{N}(t)=r N(t)$$
and explicit solution $N(t)=N_0 \exp (r t)$. The net growth rate $r$ is also called the Malthusian parameter (Thomas Malthus, 1798: Essay on the Principle of Population). Exponential growth with $r>0$ leads to population explosion and is unsustainable in Nature. With $r<0$, there is a trivial stable equilibrium at $N=0$.

The simplest model with a stable equilibrium population derives from a dynamics with immigration and death (negative growth),
$$\dot{N}(t)=c-d N(t)$$
with $c, d>0$ and explicit solution
$$N(t)=\frac{c}{d}+\left(N_0-\frac{c}{d}\right) \exp [-d t] .$$
The model describes the dynamics in a population sink. It also describes pure migration if immigration is constant and emigration occurs with a constant per-capita rate.

An alternative approach assumes that larvae (or juveniles) do not compete primarily among themselves, but with their adult conspecifics. This will be true, in particular, if adult fish eat eggs and larvae of their own species (which is indeed true for many species, where predation is purely size-dependent).

\end{prob}

We then get

$$\frac{\partial L_t(\tau)}{\partial \tau}=-m_1 L_t(\tau)-m_2 N_t L_t(\tau)$$
which simply results in
$$L_t(\Delta)=b N_t \exp \left[-\left(m_1+m_2 N_t\right) \Delta\right] .$$
We then get a so-called Ricker model for stock recruitment
$$N_{t+1}=s N_t+N_t \exp \left[r\left(1-N_t / K\right)\right]$$
with constants
$$r=\log [b]-m_1 \Delta \quad, \quad K=\frac{\log [b]-m_1 \Delta}{m_2 \Delta}$$

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