Organisms and Populations Set-1

Q1. A biologist uses the Lincoln–Petersen mark–recapture method to estimate a frog population. On the first visit she captures, marks and releases $n_1=120$ individuals. On a later visit she captures $n_2=200$ individuals, of which $m=30$ are marked. Using the estimator $N=\dfrac{n_1 n_2}{m}$, the best estimate of total population size is

Q2. Assume logistic growth given by $N(t)=\dfrac{K}{1+Ae^{-rt}}$ where $A=\dfrac{K-N_0}{N_0}$. A population has $N_0=200$, carrying capacity $K=1000$ and intrinsic growth rate $r=0.5\ \mathrm{yr^{-1}}$. How long (in years) will it take for the population to reach $N(t)=K/2$?

Q3. Given the following age-specific survivorship ($l_x$) and fecundity ($m_x$) for a species, compute the net reproductive rate $R_0=\sum l_x m_x$ and choose the correct interpretation.
Age classes and values: 0–1: $l_0=1.0,\ m_0=0$; 1–2: $l_1=0.5,\ m_1=1.6$; 2–3: $l_2=0.3,\ m_2=2.0$; 3–4: $l_3=0.15,\ m_3=0.5$.

Q4. For a population following logistic growth $\dfrac{dN}{dt}=rN\left(1-\dfrac{N}{K}\right)$ with a constant annual harvest $H$ (so $\dfrac{dN}{dt}=rN(1-N/K)-H$), the population has at least one positive equilibrium only if $H\le\dfrac{rK}{4}$. If $r=0.6\ \mathrm{yr^{-1}}$ and $K=200$ individuals, the maximum sustainable constant harvest $H_{crit}$ (individuals per year) is approximately

Q5. Consider a population whose baseline per capita growth follows $dN/dt=rN(1-N/K)$ but that exhibits a strong Allee effect at low densities (difficulty finding mates or cooperative behaviors that reduce growth at low $N$). Statement I: A critical threshold density $N_c$ exists below which the population will decline to extinction even if $K$ is large. Statement II: A strong Allee effect can produce two positive equilibria — an unstable threshold $N_c$ and a stable high-density equilibrium near $K$ — which explains why stochastic reductions below $N_c$ can lead to extinction. Which option is correct?

Q6. A biologist uses mark–recapture in a closed pond: first capture and tag $n_1=100$ fish and release them. After mixing, a sample of $n_2=80$ fish is captured, of which $m=20$ are tagged. Using $N=\dfrac{n_1 n_2}{m}$, estimate the total population size.

Q7. For a female cohort the age-specific data are: $x=0:\;l_0=1.0,\;m_0=0$; $x=1:\;l_1=0.5,\;m_1=2$; $x=2:\;l_2=0.2,\;m_2=4$; $x=3:\;l_3=0.1,\;m_3=0$. Compute $R_0=\sum l_x m_x$, $T=\dfrac{\sum x l_x m_x}{R_0}$ and estimate the intrinsic rate $r\approx\dfrac{\ln R_0}{T}$. Which value is closest to the estimated $r$ (per year)?

Q8. A population follows logistic growth with $r=0.5\ \text{month}^{-1}$ and carrying capacity $K=1000$. If current population size is $N=600$, calculate the instantaneous growth rate $dN/dt$ given $\dfrac{dN}{dt}=rN\left(1-\dfrac{N}{K}\right)$. (Individuals per month)

Q9. Two competing species A and B follow Lotka–Volterra competition with $K_A=400$, $K_B=200$, competition coefficients $\alpha_{AB}=0.5$ (effect of B on A) and $\alpha_{BA}=1.5$ (effect of A on B). Using the coexistence conditions $\alpha_{AB}<\dfrac{K_A}{K_B}$ and $\alpha_{BA}<\dfrac{K_B}{K_A}$, which outcome is most likely?

Q10. A population has $N_m=10$ breeding males and $N_f=90$ breeding females. Compute effective population size $N_e=\dfrac{4N_mN_f}{N_m+N_f}$ and the proportion of initial heterozygosity retained after 5 generations $H_5=H_0\left(1-\dfrac{1}{2N_e}\right)^5$. Which value is closest to $H_5$ expressed as a fraction of $H_0$?