Organisms and Populations Set-3

Q1. A bacterial population grows according to $N(t)=N_0e^{rt}$. If $N_0=100$ and $r=0.69\ \mathrm{h}^{-1}$, how many hours will it take to reach $N=800$?

Q2. A population follows logistic growth given by $\dfrac{dN}{dt}=rN\left(1-\dfrac{N}{K}\right)$. For $r=0.5\ \mathrm{yr}^{-1}$, $K=1000$ and initial $N_0=50$, how many years until the population first reaches $K/2$? (Use $N(t)=\dfrac{K}{1+Ae^{-rt}}$ with $A=\dfrac{K-N_0}{N_0}$.)

Q3. A species has age classes $x=0,1,2$ with survivorship and fecundity: $l_0=1.0,\ m_0=0;\ l_1=0.5,\ m_1=1;\ l_2=0.2,\ m_2=a$. Find the value of adult fecundity $a$ that gives a stable population ($R_0=\sum l_x m_x=1$).

Q4. Consider the following statements about metapopulations and habitat fragmentation.
I. A metapopulation can persist regionally even if local populations undergo frequent extinctions because empty patches can be recolonized.
II. Increasing isolation among habitat patches reduces recolonization rate and therefore increases the probability of regional extinction.

Q5. A population has age classes $x=1,2,3$ with $l_1=0.6,\ m_1=0.5;\ l_2=0.4,\ m_2=1.5;\ l_3=0.2,\ m_3=3.0$. Using $R_0=\sum l_x m_x$, mean generation time $T=\dfrac{\sum x l_x m_x}{R_0}$ and $r\approx\dfrac{\ln R_0}{T}$, estimate the doubling time $t_d\approx\dfrac{\ln 2}{r}$. Choose the closest value.

Q6. In a closed pond a capture–recapture study used the Lincoln–Petersen estimator $N=\dfrac{n_1 n_2}{m_2}$. Researchers first marked and released $n_1=80$ fish. Later they captured $n_2=40$ fish, of which $m_2=16$ were marked. What is the best estimate of the total fish population $N$?

Q7. A population follows continuous logistic growth with $N(t)=\dfrac{K}{1+Ae^{-rt}}$, where $A=\dfrac{K-N_0}{N_0}$. For $r=0.5\ \text{yr}^{-1}$, $K=1000$ and initial $N_0=50$, estimate the time (in years) required for $N(t)$ to reach $0.9K$ (90% of $K$).

Q8. Given age-specific data $x=0,1,2$ with survivorship $l_x=1.0,\,0.6,\,0.2$ and fecundity $m_x=0,\,1.5,\,2.0$, compute $R_0=\sum l_x m_x$, generation time $T=\dfrac{\sum x l_x m_x}{R_0}$ and approximate intrinsic rate $r\approx \dfrac{\ln R_0}{T}$. Which value is closest to $r$ (per time unit)?

Q9. A harvested population follows $\dfrac{dN}{dt}=rN\left(1-\dfrac{N}{K}\right)-H$ with $r=0.6\ \text{yr}^{-1}$, $K=1000$ and constant harvest $H=50$ individuals per year. Equilibria satisfy $rN(1-N/K)=H$. If the starting population is $N_0=100$, what is the long-term outcome?

Q10. Field observations give: when $N_t=50$, $N_{t+1}=75$; when $N_t=150$, $N_{t+1}=180$; when $N_t=200$, $N_{t+1}=210$. Assuming discrete logistic dynamics approximated by per capita growth
$g(N)=\dfrac{N_{t+1}-N_t}{N_t}=r-\dfrac{r}{K}N_t$, use these data to estimate the carrying capacity $K$ (nearest whole number).