I have tried searching online but I don't seem to hit the right keyword to get an answer. Here is a 2nd order Problem about plants:
Plants germinate in spring, bloom in summer and produce seeds in early fall. A fraction of seeds survive in winter and others get eaten or rotten. After that, a fraction of seeds germinate in next spring and produce more seeds. However not all seeds germinate in the same spring, some of them germinate in the next spring or do not survive the winter because they get eaten or rotten.
The model parameters are
$p(n)$ plants in year $n$
$r$ - number of seeds produced per plant
$s$ - fraction of seeds that survived in the winter
$a$ - fraction of one year old seeds that can germinate
$b$ - fraction of two year old seeds that can germinate
Now according to my notes, the 2nd order system that corresponds to the problem is $$ p(n)= p(n-1)rsa + p(n-2)rs(1-a)sb $$ I do not understand how $rsa$ describe $p(n-1)$ and $p(n-2)$ it just looks like everything was multiplied out. Are there any resources I can study on how to make 2nd order systems from mathematical models?
Does anyone have any comments or ideas how $p(n-1)rsa$ and $p(n-2)rs(1-a)sb$ was derived?
If you have 100 plants, each plant produces five seeds, but only three-fifths of the seeds survive the winter, and one-half of them actually germinate, how many plants will you have next year?
Now if you have $P_n$ plants, each plant produces $r$ seeds, but only $s$ of the seeds survive the winter, and $a$ of them actually germinate, how many plants will you have next year?