# 다은이는 여기 글 써

###### tags: `수리생물학`

## 1.6 An example: Leslie’s Age-Structured Model
### Leslie's Age-Structured Model
The Leslie matrix model is a linear, first-order system of difference equations that models the dynamics of age-structured populations.

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### What is an age-structured model?
1. One of many types of structured models
2. Structure : an organization of division of the population into various parts such as age, size, or stage.
3. Stage-structured model에서 몇 개의 발달상의 단계
Ex) 곤충:알,애벌레,번데기,성충
4. Age-structured model은 stage-structured model에서 개체군을 다시 연령그룹으로 나눈것
5. In human demography, 연령그룹을 0-5,5-10세 등 구간을 5년으로 나눕니다. 개체들은 size에 따라서 그룹화 된다.
6. 단계들 사이에서 Dynamic interactions은 연령 또는 크기는 개체군 구조가 시간에 걸쳐 어떻게 변화 되는지를 결정함.
7. 개체군을 구조화하는 다른 방법은, 연령,크기,단계에 더하여 성과공간구조들을 포함함.

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### Leslie's Matrix
Assume the population is closed to migration and only the females are modeled.

Males are present, but are not specifically modeled.

Ratio of males to females = a:b

The survival rate per age group is the same for males and females,
then the number of males equals the nunber of females times a/b.

Total number of age groups = m

During the interval of time t to t+1, individuals ‘age’ from i to i+1

$\ x_{t}$=the number of females in the ith age group at time t.
$\ 𝑏_{𝑖}$ = the average number of newborn females produced by one female in the ith age group that survive through the time interval in which they were born.
$\ 𝑠_{𝑖}$ = the fraction of the ith age group that live to the (i+1)st age 0 < $\ 𝑠_{𝑖}$ $\le$ 1

첫번째 연령그룹은
$\ 𝑥_1(t+1)=𝑏_1 𝑥_1(t)+𝑏_2 𝑥_2(t)+⋯+𝑏_𝑚 𝑥_𝑚(t)=∑b_1x_1(t)$
$\ x_{𝑖+1}(t+1)= 𝑠_𝑖 𝑥_𝑖(t)$