Biodiversity and Conservation Set-5

Q1. In a community with four species having abundances 10, 20, 30 and 40 individuals respectively, calculate Simpson's diversity index using $D = 1 - \sum p_i^2$ where $p_i$ is the proportion of species $i$. (Take proportions to two decimal places if needed.)

Q2. Two communities each have five species. Community A has abundances (40, 30, 10, 10, 10) and Community B has abundances (96, 1, 1, 1, 1). Using Simpson's index $D = 1 - \sum p_i^2$, which statement is correct about their diversities?

Q3. For a population with $N_m=40$ breeding males and $N_f=10$ breeding females, effective population size is given by $N_e=\dfrac{4N_mN_f}{N_m+N_f}$ and the approximate per-generation increase in inbreeding coefficient is $\Delta F \approx \dfrac{1}{2N_e}$. Calculate $\Delta F$ (rounded to four decimal places).

Q4. Using the species–area relationship $S = cA^z$, consider a fixed total area $2A$. Compare species richness of (i) one reserve of area $2A$ and (ii) two separate reserves each of area $A$ (assume species counts in the two small reserves are additive, i.e., no overlap). For which values of the exponent $z$ will the single large reserve support strictly more species than the two small reserves?

Q5. Under Wright's island model, $F_{ST} \approx \dfrac{1}{1 + 4N_e m}$, where $N_e$ is effective population size and $m$ is the migration rate per generation. For a metapopulation with $N_e = 50$, what minimum migration rate $m$ (as a fraction of the population per generation) is required to keep $F_{ST} < 0.10$?

Q6. In a small isolated population there are 10 breeding males and 30 breeding females. Using the formula $N_e=\dfrac{4N_mN_f}{N_m+N_f}$, calculate the effective population size $N_e$.

Q7. A population has effective size $N_e=50$. Expected heterozygosity after $t$ generations is given by $H_t=H_0\left(1-\dfrac{1}{2N_e}\right)^t$. Approximately after how many generations will heterozygosity decline to half its initial value ($H_t=0.5H_0$)?

Q8. A continuous forest of area $10\,000\ \text{km}^2$ is reduced to $2\,500\ \text{km}^2$ due to habitat loss. Using the species–area relationship $S=cA^z$ with $z=0.25$ (and $c$ unchanged), what percentage of the original species richness is expected to remain?

Q9. Assertion (A): Fragmenting a population into several small isolated reserves of equal total area increases the overall effective population size $N_e$ compared to a single contiguous reserve of the same total area.
Reason (R): Isolation of subpopulations increases genetic drift and inbreeding, which generally reduces $N_e$ and genetic variability.

Q10. Two reserves, each of area $1\ \text{km}^2$, are designed: one circular and one square. Edge effects penetrate $w=100\ \text{m}$ into the periphery. Using $A=\pi r^2$ for the circle and $A=s^2$ for the square, which shape retains a larger proportion of core habitat and what are the approximate core-area percentages for each (use $\pi\approx3.1416$)?

Q11. A habitat follows the species–area relation $S=cA^{z}$ with $z=0.25$. If the habitat area is reduced to one-sixteenth of its original value ($A_2=\tfrac{A_1}{16}$) due to fragmentation, approximately what percentage of the original species richness is expected to persist?

Q12. In a small mammal population there are $N_m=12$ breeding males and $N_f=48$ breeding females. Using the effective population size formula $N_e=\dfrac{4N_mN_f}{N_m+N_f}$, compute $N_e$ and compare the expected rate of genetic drift to an ideal population whose census size equals the total number of breeders ($N=60$). Which statement is correct?

Q13. In a metapopulation model the equilibrium fraction of occupied patches is $f^{*}=1-\dfrac{e}{c}$, where $e$ is patch extinction rate and $c$ the colonization rate. Initially $e=0.2\ \mathrm{yr^{-1}}$ and $c=0.6\ \mathrm{yr^{-1}}$. Habitat fragmentation causes $e$ to increase by 50% and $c$ to decrease by 25%. What fraction of the original equilibrium occupancy $f^{*}$ remains after fragmentation?

Q14. Two reserves have equal total area $A=100\ \mathrm{km^{2}}$. Reserve I is circular; Reserve II is square. Edge effects penetrate inward to a width $w=1\ \mathrm{km}$ and make the edge zone unsuitable; the remaining area is the ‘core’. Using $r=\sqrt{A/\pi}$ for the circle and $s=\sqrt{A}$ for the square, which reserve retains the larger core area and what are the approximate core areas?

Q15. Consider a community with species abundances S1=50, S2=30, S3=15, S4=5 (total $N=100$). Shannon diversity is $H'=-\sum p_i\ln p_i$ where $p_i$ are relative abundances. Two events occur separately: (i) species S4 (5 individuals) is extirpated; (ii) twenty individuals of the dominant species S1 are removed, yielding new abundances S1=30, S2=30, S3=15, S4=5 (total $N=80$). Which scenario causes a larger decrease in $H'$ relative to the initial community?