Question Solved1 Answer 4. Consider the dynamics of a population that consists of \( n \) sub-populations, with their respective sizes at time \( t \) represented by \( X_{1}(t), X_{2}(t), \cdots, X_{n}(t) \). Let us assume that for each sub-population, \[ \frac{\mathrm{d} X_{1}(t)}{\mathrm{d} t}=r_{1} X_{1}, \frac{\mathrm{d} X_{2}(t)}{\mathrm{d} t}=r_{2} X_{2}, \cdots, \frac{\mathrm{d} X_{n}(t)}{\mathrm{d} t}=r_{n} X_{n} . \] So the dynamics of these \( n \) sub-poluations are completely independent from each other, and each is growing or decaying "exponentially" depending on the sign of its \( r \). Now introducing the per capita growth rate (PCGR) of the total population at time \( t \) : \[ \bar{r}(t)=\frac{1}{X_{\text {tot }}(t)} \frac{\mathrm{d} X_{\text {tot }}(t)}{\mathrm{d} t} \text {, where } X_{\text {tot }}(t)=\sum_{i=1}^{n} X_{i}(t) \text {. } \] (a) The PCGR of the total population is not a constant over time. Show that it is the "average" of the PCGRs of the sub-populations weighted by the population: \[ \bar{r}=\frac{\sum_{i=1}^{n} X_{i} r_{i}}{\sum_{i=1}^{n} X_{i} .} \] (b) More interestingly, show that \[ \frac{\mathrm{d}}{\mathrm{d} t} \bar{r}(t)=\frac{\sum_{i=1}^{n} X_{i}\left(r_{i}-\bar{r}(t)\right)^{2}}{\sum_{i=1}^{n} X_{i}(t)} \geq 0 . \] (c) For \( n=2 \), show that \[ \frac{\mathrm{d}}{\mathrm{d} t}\left(\frac{X_{1}}{X_{1}+X_{2}}\right)=\frac{\left(r_{1}-r_{2}\right) X_{1} X_{2}}{\left(X_{1}+X_{2}\right)^{2}} . \] Based on this mathematical expression, discuss what happens to the two sub-populations if \( r_{1}>r_{2} \), and if \( r_{2}>r_{1} \) ? What general conclusions can you reach for the total population \( X_{1}(t)+X_{2}(t) \) ? (d) In the biological context, the per capita growth rate is often used as the fitness of a population. Discuss the mathematical result in \( (b) \) with respect to the statement that "The fitness of a population always increases when there are variations within the population."

BUS7A5 The Asker · Probability and Statistics

Transcribed Image Text: 4. Consider the dynamics of a population that consists of \( n \) sub-populations, with their respective sizes at time \( t \) represented by \( X_{1}(t), X_{2}(t), \cdots, X_{n}(t) \). Let us assume that for each sub-population, \[ \frac{\mathrm{d} X_{1}(t)}{\mathrm{d} t}=r_{1} X_{1}, \frac{\mathrm{d} X_{2}(t)}{\mathrm{d} t}=r_{2} X_{2}, \cdots, \frac{\mathrm{d} X_{n}(t)}{\mathrm{d} t}=r_{n} X_{n} . \] So the dynamics of these \( n \) sub-poluations are completely independent from each other, and each is growing or decaying "exponentially" depending on the sign of its \( r \). Now introducing the per capita growth rate (PCGR) of the total population at time \( t \) : \[ \bar{r}(t)=\frac{1}{X_{\text {tot }}(t)} \frac{\mathrm{d} X_{\text {tot }}(t)}{\mathrm{d} t} \text {, where } X_{\text {tot }}(t)=\sum_{i=1}^{n} X_{i}(t) \text {. } \] (a) The PCGR of the total population is not a constant over time. Show that it is the "average" of the PCGRs of the sub-populations weighted by the population: \[ \bar{r}=\frac{\sum_{i=1}^{n} X_{i} r_{i}}{\sum_{i=1}^{n} X_{i} .} \] (b) More interestingly, show that \[ \frac{\mathrm{d}}{\mathrm{d} t} \bar{r}(t)=\frac{\sum_{i=1}^{n} X_{i}\left(r_{i}-\bar{r}(t)\right)^{2}}{\sum_{i=1}^{n} X_{i}(t)} \geq 0 . \] (c) For \( n=2 \), show that \[ \frac{\mathrm{d}}{\mathrm{d} t}\left(\frac{X_{1}}{X_{1}+X_{2}}\right)=\frac{\left(r_{1}-r_{2}\right) X_{1} X_{2}}{\left(X_{1}+X_{2}\right)^{2}} . \] Based on this mathematical expression, discuss what happens to the two sub-populations if \( r_{1}>r_{2} \), and if \( r_{2}>r_{1} \) ? What general conclusions can you reach for the total population \( X_{1}(t)+X_{2}(t) \) ? (d) In the biological context, the per capita growth rate is often used as the fitness of a population. Discuss the mathematical result in \( (b) \) with respect to the statement that "The fitness of a population always increases when there are variations within the population."
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Transcribed Image Text: 4. Consider the dynamics of a population that consists of \( n \) sub-populations, with their respective sizes at time \( t \) represented by \( X_{1}(t), X_{2}(t), \cdots, X_{n}(t) \). Let us assume that for each sub-population, \[ \frac{\mathrm{d} X_{1}(t)}{\mathrm{d} t}=r_{1} X_{1}, \frac{\mathrm{d} X_{2}(t)}{\mathrm{d} t}=r_{2} X_{2}, \cdots, \frac{\mathrm{d} X_{n}(t)}{\mathrm{d} t}=r_{n} X_{n} . \] So the dynamics of these \( n \) sub-poluations are completely independent from each other, and each is growing or decaying "exponentially" depending on the sign of its \( r \). Now introducing the per capita growth rate (PCGR) of the total population at time \( t \) : \[ \bar{r}(t)=\frac{1}{X_{\text {tot }}(t)} \frac{\mathrm{d} X_{\text {tot }}(t)}{\mathrm{d} t} \text {, where } X_{\text {tot }}(t)=\sum_{i=1}^{n} X_{i}(t) \text {. } \] (a) The PCGR of the total population is not a constant over time. Show that it is the "average" of the PCGRs of the sub-populations weighted by the population: \[ \bar{r}=\frac{\sum_{i=1}^{n} X_{i} r_{i}}{\sum_{i=1}^{n} X_{i} .} \] (b) More interestingly, show that \[ \frac{\mathrm{d}}{\mathrm{d} t} \bar{r}(t)=\frac{\sum_{i=1}^{n} X_{i}\left(r_{i}-\bar{r}(t)\right)^{2}}{\sum_{i=1}^{n} X_{i}(t)} \geq 0 . \] (c) For \( n=2 \), show that \[ \frac{\mathrm{d}}{\mathrm{d} t}\left(\frac{X_{1}}{X_{1}+X_{2}}\right)=\frac{\left(r_{1}-r_{2}\right) X_{1} X_{2}}{\left(X_{1}+X_{2}\right)^{2}} . \] Based on this mathematical expression, discuss what happens to the two sub-populations if \( r_{1}>r_{2} \), and if \( r_{2}>r_{1} \) ? What general conclusions can you reach for the total population \( X_{1}(t)+X_{2}(t) \) ? (d) In the biological context, the per capita growth rate is often used as the fitness of a population. Discuss the mathematical result in \( (b) \) with respect to the statement that "The fitness of a population always increases when there are variations within the population."
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