1. SETTINGS 1.1. Assume that a group of same kind of birds live in a place. The

Question

1. SETTINGS 1.1. Assume that a group of same kind of birds live in a place. The population of birds of age i is denoted by ni. Remark 1. Here n actually refers to the population of female birds of age i. In this article we only count female birds. So all populations are referred to the populations of female birds. We omit the gender description just for simplicity 1.2. The birds can grow up to w years old. As soon as they reach age w, they will die. So n 0. So are all later n's 1.3. At each age they might meet some dangers and have a possibility to die. Thus we can consider the survive rate. Let s be the survive rate which describes the fraction of individuals that survives from age class i to age class i 1. since all age w -1 birds will become age w and die, su-1 0. So are all later s's 1.4. Birds can reproduce. Let fi be the fecundity, which is the per capita average number of offspring reaching no stage born from mother of the age class i. Note that when measuring the number fi, the survive rate is already considered. 1.5. To sum up, at age i, there are ni birds. Among them only sini can make to the next age. But they will also reproduce baby birds. There will be total fini baby birds making to the no stage (after considering the survive rate for new born birds). 1.6. When we talk about age classes from different years we use subindex. For example Init refers to the number of birds of age i in year t.

Explanation / Answer

5.1The linear space we are interested in here is the space spanned by the vectors nt ,

here nt = [ n0 n1    n2    n3 .......... nw-1 ]Tt   .   w = average life span of a bird .,

T means transpose of the vector matrix

n0 = no. of birds alive and existing , of age 0 i.e. newly born at time instant t .

  n1 = no. of birds alive and existing , of age 1 at time instant t .

  n2 = no. of birds alive and existing , of age 2 at time instant t .

.................

................

  nw-1 = no. of birds alive and existing , of age w-1 at time instant t .

5.2 The linear transformation we are interested in is nt ---> nt+1 ., i.e. given the value/state of no. of birds of different ages at time t ,to find the value/state of no. of birds of different ages at time t+1 .

  

and this linear transformation is accomplished by the transformation matrix L (Lesslie matrix here) .

  nt X L ---> nt+1

5.3 If eigenvalue of the transformation matrix L is 1 associated to L's eigen vector nt , this shows that value of vector nt has stabilized and will not change . i.e. nt+1 will be approx. same as nt .

The interpretation is that the no. of birds for any age group is not going to change over the years .

nt *L = nt+1 .

5.4 Eigenvalues greater than 1 would mean no of birds in any age class increasing scalarly , Eigenvector stabilized , it means

i.e. ni at time t+1 = K.( ni at time t) here ni is no. of birds in age class i

K is a constant .

i.e. nt+1 = K*nt . POPULATION OF BIRDS IN ALL AGE CLASSES INCREASING SCALARLY .

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