en:ecovirt:roteiro:den_ind:di_base
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en:ecovirt:roteiro:den_ind:di_base [2017/09/12 16:23] melina.leite [Exercises: playing in the fields of the Lord] |
en:ecovirt:roteiro:den_ind:di_base [2022/09/19 13:16] 127.0.0.1 external edit |
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| The numbers of deaths and births are the result of //per capita// rates multiplied by population size: | The numbers of deaths and births are the result of //per capita// rates multiplied by population size: | ||
| - | * $ B=bN_t $ | + | $$ B=bN_t $$ |
| - | * $ D=dN_t $ | + | $$ D=dN_t $$ |
| where: $b$ = //per capita// birth rate at each generation; $d$ = //per capita// death rate at each generation. Note that the rate does not change with the size of the population, and the number of births and deaths is proportional to the population size. Let's just clarify one more premise, for educational purposes: births and deaths occur simultaneously in the population at the time interval $t$. We can then say that: | where: $b$ = //per capita// birth rate at each generation; $d$ = //per capita// death rate at each generation. Note that the rate does not change with the size of the population, and the number of births and deaths is proportional to the population size. Let's just clarify one more premise, for educational purposes: births and deaths occur simultaneously in the population at the time interval $t$. We can then say that: | ||
| - | * $N_{t+1} = N_t + bN_t-dN_t $ | + | $$N_{t+1} = N_t + bN_t-dN_t $$ |
| - | * $N_{t+1} = N_t + (b-d)N_t $ | + | $$N_{t+1} = N_t + (b-d)N_t $$ |
| if we define a factor of discrete growth: $r_t = b-d$ | if we define a factor of discrete growth: $r_t = b-d$ | ||
| - | * $N_{t+1} = (1+r_t)N_t$ | + | $$N_{t+1} = (1+r_t)N_t$$ |
| - | * $\frac{N_{t+1}}{N_t} = 1+r_t$ | + | $$\frac{N_{t+1}}{N_t} = 1+r_t$$ |
| As $1 + rt$ is a constant, let's designate it as $\lambda$, a positive number that expresses the proportional increase of the population from one generation to another. Therefore: | As $1 + rt$ is a constant, let's designate it as $\lambda$, a positive number that expresses the proportional increase of the population from one generation to another. Therefore: | ||
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| We can also project the population for other generations, using iterations: | We can also project the population for other generations, using iterations: | ||
| - | * $N_{t+2} = 105 \times 1,05 = 110,25$ | + | $$N_{t+2} = 105 \times 1,05 = 110,25$$ |
| - | * $N_{t+3} = 110,25 \times 1,05 = 115,7625$ | + | $$N_{t+3} = 110,25 \times 1,05 = 115,7625$$ |
| continuing and taking the population size at zero time ($N_0$): | continuing and taking the population size at zero time ($N_0$): | ||
| - | * $N_{t+4}= N_0 \times \lambda \times \lambda \times \lambda \times \lambda$ | + | $$N_{t+4}= N_0 \times \lambda \times \lambda \times \lambda \times \lambda$$ |
| - | * $N_{t+4}= N_0 \lambda^4 $ | + | $$N_{t+4}= N_0 \lambda^4 $$ |
| Generalizing: | Generalizing: | ||
en/ecovirt/roteiro/den_ind/di_base.txt · Last modified: 2022/09/19 13:20 (external edit)
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