Organisms and Populations Set-4

Q1. In a pond study 40 fish were captured, marked and released. Later a sample of 60 fish was taken and 10 of them were found to be marked. Using the mark–recapture estimator $N=\dfrac{sn}{x}$, the best estimate of the population size is:

Q2. A population follows logistic growth given by $\dfrac{dN}{dt}=rN\left(1-\dfrac{N}{K}\right)$. For $r=0.5\ \text{yr}^{-1}$, $K=1000$ individuals and current $N=600$, which statement is correct?

Q3. Given this life-table data for a species with age classes 0–1, 1–2 and 2–3 years: survivorship $l_0=1$, $l_1=0.6$, $l_2=0.3$ and fecundities $m_0=0$, $m_1=1.2$, $m_2=2.5$. Calculate the net reproductive rate $R_0=\sum l_x m_x$ and state the likely population trend.

Q4. Assertion (A): In Levins' metapopulation model with per-patch colonization rate $c$ and extinction rate $e$, the equilibrium fraction of occupied patches is $p^*=1-\dfrac{e}{c}$ (for $c>e$).
Reason (R): If fragmentation halves the colonization rate $c$ (to $c/2$) while $e$ remains unchanged, the equilibrium occupancy $p^*$ will be halved.

Q5. A population with logistic growth $\dfrac{dN}{dt}=rN\left(1-\dfrac{N}{K}\right)$ experiences occasional sharp declines due to environmental variability. Two harvesting strategies remove the same long-term average biomass: (i) a fixed quota equal to the maximum sustainable yield $H=\dfrac{rK}{4}$; (ii) a proportional harvest at rate $h$ chosen so the long-term mean removal equals that from (i). Which harvesting strategy will generally reduce extinction risk during population dips?

Q6. A bacterial culture grows exponentially with intrinsic rate of increase $r=0.6\ \text{h}^{-1}$. If the initial population is $N_0=100$, what is the population after $3$ hours? (Use $N_t=N_0e^{rt}$)

Q7. In a mark–recapture study, $M=120$ individuals were marked and released. Before recapture, $80$ newborn (unmarked) individuals were added to the population (no deaths or emigration). In the recapture sample of size $n=60$, $r=10$ marked individuals were found. Using the estimator $N=\dfrac{Mn}{r}$ for the population at the time of recapture, what was the population size immediately after the initial marking (i.e., before the 80 births)?

Q8. Two populations of the same species have identical net reproductive rate $R_0=4.0$. Population P has mean generation time $T_P=2\ \text{yr}$ and population Q has $T_Q=3\ \text{yr}$. Using the approximation $r\approx\dfrac{\ln R_0}{T}$ for intrinsic rate $r$, which statement is correct?

Q9. Assertion (A): In populations with a strong Allee effect, there can be a critical density threshold below which the population declines to extinction even though carrying capacity $K$ is large.
Reason (R): A modified per-capita growth term may include a factor like $\left(\dfrac{N}{A}-1\right)$ (with Allee constant $A$); when $N<A$ that factor becomes negative, making per-capita growth negative and causing small populations to shrink.

Q10. Two species compete according to the Lotka–Volterra competition model: $\dfrac{dN_1}{dt}=r_1N_1\left(1-\dfrac{N_1+\alpha N_2}{K_1}\right)$ and $\dfrac{dN_2}{dt}=r_2N_2\left(1-\dfrac{N_2+\beta N_1}{K_2}\right)$. Given $K_1=500,\ K_2=300,\ \alpha=0.6,\ \beta=0.9$ (with $r_1,r_2>0$), which outcome is predicted?

Q11. A bacterial population has initial size $N_0=200$ and grows continuously with intrinsic rate $r=0.2\ \text{day}^{-1}$. Using $N_t=N_0e^{rt}$, what is the population size after $t=3$ days (approximate)?

Q12. A population follows logistic growth $\dfrac{dN}{dt}=rN\left(1-\dfrac{N}{K}\right)$ with $r=1\ \text{yr}^{-1}$ and $K=500$. If a constant harvest $H=100$ individuals yr$^{-1}$ is removed, equilibrium sizes $N^*$ satisfy $rN^*(1-N^*/K)=H$. Which pair (rounded) gives the non-negative equilibria?

Q13. Consider an Allee-effect model $\dfrac{dN}{dt}=rN\!\left(1-\dfrac{N}{K}\right)\!\left(\dfrac{N}{A}-1\right)$ with $r=1\ \text{yr}^{-1}$, $K=1000$, $A=50$. If initial population $N(0)=30$, which outcome is expected?

Q14. A species has age classes $x=0,1,2$ yr with survivorship $l_x=1.0,\ 0.6,\ 0.3$ and fecundity $m_x=0,\ 1.5,\ 3.0$ respectively. Using $R_0=\sum l_xm_x$, $T=\dfrac{\sum x l_x m_x}{R_0}$ and the approximation $r\approx \dfrac{\ln R_0}{T}$, what is the approximate intrinsic rate of increase $r$ (yr$^{-1}$)? Choose the closest value.

Q15. Two species compete according to Lotka–Volterra competition: $\dfrac{dN_1}{dt}=r_1N_1\left(1-\dfrac{N_1+\alpha N_2}{K_1}\right)$, $\dfrac{dN_2}{dt}=r_2N_2\left(1-\dfrac{N_2+\beta N_1}{K_2}\right)$. For $r_1=r_2=1\ \text{yr}^{-1}$, $K_1=500,\ K_2=300,\ \alpha=1.2,\ \beta=0.5$, what is the long-term outcome?