Animal survival and renewal An animal population currently...

Animal survival and renewal An animal population currently has 7400 members and is reproducing at the rate $R(t)=2240+60 t$ members/year. The proportion of members that survive after $t$ years is given by $S(t)=1 /(t+1)$ . (a) How many of the original members survive four years? (b) How many new members are added during the next four years? (c) Explain why the animal population four years from now is not the same as the sum of your answers from parts (a) and (b).

Animal survival and renewal An animal population currently has 7400 members and is reproducing at the rate $R(t)=2240+60 t$ members/year. The proportion of members that survive after $t$ years is given by $S(t)=1 /(t+1)$ .
(a) How many of the original members survive four years?
(b) How many new members are added during the next four years?
(c) Explain why the animal population four years from now is not the same as the sum of your answers from parts (a) and (b).

Key Concepts

Overlapping Generations and Cohort Effects

Overlapping generations occur when individuals of different age groups coexist and contribute to the overall population dynamics. In such scenarios, the original population and the new cohorts produced during a time period are not isolated; their contributions are interdependent because the survival and reproduction rates apply continuously and variably over time, meaning that the total future population is derived from an integrated effect rather than a simple sum of separate groups.

Integration in Continuous Population Models

In continuous population dynamics, integration is used to accumulate the contributions of births and their survival over a time interval. Since individuals are born at different times and experience different survival probabilities between their time of birth and the target time, integrating the birth rate multiplied by the corresponding survival probability over time gives the number of new individuals alive at the later date.

Survival Function

A survival function describes the probability that an individual survives up to a given time. In population models, it is used to adjust the number of individuals present at an earlier time to reflect how many are expected to be alive at a later time, based on a time?dependent mortality rate.

Reproduction Rate Function

The reproduction rate function specifies how many new individuals are produced per unit time. It is often a function of time and accounts for the varying rates at which individuals in the population generate offspring, allowing for the modeling of continuous or non?constant birth processes.

Transcript

00:01 Okay, to start, i've written down a couple notes here.

00:04 First, we're given that we start with 7 ,400 members, and we're given an equation that the population is reproducing at a rate of r of t equals 2 ,240 plus 60 t members per year, and that the proportion of members that survive after two years is given by s of t equals 1 over t plus 1.

00:27 So in part a, it asks us to find how many of the original members survive after four years.

00:40 So we're going to plug this into the s of t equals 1 over t plus 1.

00:46 We're going to plug in t equals 4.

00:48 So we get s of 4 equals 1 over 4 plus 1, which is going to be equal to 1 5th.

01:09 And then, so one -fifth of our members will survive after four years.

01:16 So then one -fifth times our original 7 ,400 members gives us a number of, if we plug this into a calculator real quick, of 1 ,480 members.

01:51 And that's for part a.

01:57 Okay, so for part b, it asks us to find how many new members are added during the next four years.

02:06 Now we know that this equation gives us the rate at which the population is reproducing.

02:16 However, we don't want the rate we want the number.

02:18 So this is where we're going to use a definite integral from 0 to 4 of r of 2.

02:26 So we have the definite integral from 0 for 2 ,240 plus 60t, dt...

Please give Ace some feedback