Q.(a)

Exponential Growth Rate

A quantity grows exponentially when its rate of change is proportional to its current size: the more there is, the faster it grows. This differs sharply from linear growth, where a fixed amount is added each step. In exponential growth the quantity multiplies by the same factor over equal time intervals.

The differential equation

Let $y(t)$ be the quantity and $k > 0$ the proportionality constant. The rate law "rate of change proportional to the current amount" becomes

This is a separable equation. Integrating,

where $y_0 = y(0)$ is the starting value. The constant $k$ is the growth rate: a larger $k$ means faster growth. (If $k < 0$, the very same equation describes exponential decay.)

Reading the growth rate

Over each unit of time, $y$ is multiplied by $e^{k}$. So if the quantity doubles every unit of time, then $e^{k} = 2$, giving $k = \log 2$. This is how a stated doubling time is converted into the constant $k$.

Where it appears …

The Species Area Relationship: A First Look

Imagine you are walking through a small park near your home. You might spot a few birds, some insects, and a handful of plant species. Now imagine that same walk through a large forest reserve — hundreds of times bigger. Would you expect to see more kinds of birds, more types of insects, more varieties of trees? Almost certainly yes. That simple, intuitive observation is the seed of the Species Area Relationship.

What It Means

The Species Area Relationship (often abbreviated as SAR) is a pattern ecologists have observed across the natural world: as the area you sample increases, the number of species you find also increases. It is not a vague guess — it is a consistent, well-documented relationship that holds true for most groups of organisms, from plants and birds to insects and mammals.

Why does this happen? A larger area typically contains more habitats — forests, grasslands, wetlands, rocky outcrops — and each habitat supports its own set of species. A bigger area also tends to have more individuals, and with more individuals you are more likely to encounter rare species that might be absent from a small patch. In short, area acts as a rough proxy for ecological diversity and complexity.

Key Points to Remember

• The relationship is positive: bigger area → more species.
• It is not linear — doubling the area does not double the number of species. The increase slows down as area gets very large.
• The pattern holds across scales: from a single leaf (hosting tiny insects and fungi) to an entire continent.

Why It Matters

The Species Area Relationship is not just an academic curiosity. It has real-world consequences, especially for conservation.

• Designing protected areas: If you want to preserve a certain number of species, you need to know how much area is required. A small reserve may protect only a fraction of the region's biodiversity.
• Predicting extinctions: When a habitat is destroyed or fragmented, the remaining area shrinks. Using the SAR, ecologists can estimate how many species are likely to be lost as a result.
• Understanding island biology: The relationship was first studied on islands, where area is clearly defined and isolation limits immigration. The same logic applies to "habitat islands" — patches of forest surrounded by farmland, or national parks surrounded by cities. …