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Propagation of Annual Plants

Propagation of Annual Plants

We will become acquainted with some of the submissions of difference equation in this chapter. We give the mathematical form of diverse popular phenomenon in period of distinction equation. The answer of some of these model will be considered in feature in section 4.

2.1 Propagation of Annual Plants

In this part, our target here is to evolve a mathematical model that recounts the number of plants in any yearned generation[s.Elayd said]” that plants make kernels at the end of their development time of the year (say August), after which they die. Furthermore, only a part of these kernels endure the winter, and those that endure germinate at the starting of the time of the year (say May)”, giving increase to a new lifetime of plants.

Let

? = number of seeds produced per plant in August,

a= fraction of one-year-old seeds that germinate in May,

ß = fraction of two-year-old seeds that germinate in May,

?= fraction of seeds that survive a given winter.

If p(n) denotes the number of plants in generation n, then

p(n) = +

p(n) = a s1(n) + ß s2(n) (2.1)

where s1(n) (respectively, s2(n)) is the number of one-year-old (two-year

old) seeds in April (before germination). Observe that the number of seeds

left after germination may be written as

seeds left=

This gives rise to two equations:

~ s1(n) = (1 - a)s1(n) (2.2)

~ s2(n) = (1 - ß)s2(n); (2.3)

~ ~

where s1(n) (respectively, s2(n)) is the number of one-year (two-year-old)

seeds left in May after some have germinated.

New seeds s0(n) (0-year-old)are produced in August at the rate of ? per plant,

s0(n) = ? p(n): (2.4)

After winter, seeds s0(n) that were new in generation n will be one year old

in the next generation n + 1, and a fraction ? s0(n) of them will survive.

Hence

s1(n + 1) = ? s0(n);

or, by using formula (2.4), we have

s1(n + 1) = ? ? p(n): (2.5)

Similarly,

~ s2(n + 1) = ? s1(n);

which yields, by formula (2.2),

s2(n + 1) = ? (1 - a)s1(n);(2.6)

s2(n + 1) = (1 - a)p(n -1): (2.7)

Substituting for s1(n + 1), s2(n + 1) in expressions (2.5) and (2.7) into formula

(2.1)Gives

p(n + 1) = ? ? a p(n) + ? ß (1 - a )p(n -1);(2.8)

or

p(n + 2) = ? ? a p(n + 1) + ? ß (1 - a )p(n): (2.9)

Which is the difference equation model for the Propagation of Annual Plants.

2.2 Gamblers Ruin

Consider a game that gives a probability q of winning 1 dollar and a probability 1? q of losing 1 dollar where 0 =q = 1. If a player begins with 10 dollars, and intends to play the game repeatedly until he either goes broke or increases his holdings to N dollars Let p(n) denote the probability that the gambler will be ruined.This He may be ruined in two ways. First, winning the next game; the probability of this event is q; then his fortune will be n + 1,and the probability of being ruined will become ...

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