Question

One model for plant competition assumes that there is a zone of resource depletion around each plant seedling. Depending on the size of the zones and the density of the plants, the zones of resource depletion may overlap with those of other seedlings in the vicinity. When the seeds are randomly dispersed over a wide area, the number of neighbors that a seedling may have usually follows a Poisson distribution with a mean equal to the density of seedlings per unit area. Suppose that the density of seedlings is four per square meter \(\left(\mathrm{m}^{2}\right)\). a. What is the probability that a given seedling has no neighbors within \(1 \mathrm{~m}^{2} ?\) b. What is the probability that a seedling has at most three neighbors per \(\mathrm{m}^{2}\) ? c. What is the probability that a seedling has five or more neighbors per \(\mathrm{m}^{2} ?\) d. Use the fact that the mean and variance of a Poisson random variable are equal to find the proportion of neighbors that would fall into the interval \(\mu \pm 2 \sigma .\) Comment on this result.

Short answer

Question: Calculate the probabilities for the following cases of seedlings with a Poisson distribution in a square meter: a. No neighbors. b. At most 3 neighbors. c. 5 or more neighbors. d. Proportion of neighbors in the interval \(\mu±2\sigma\). Answer: a. The probability of no neighbors in a square meter is approximately 1.83%. b. The probability of at most 3 neighbors in a square meter is approximately 43.35%. c. The probability of 5 or more neighbors in a square meter is approximately 37.12%. d. The proportion of neighbors in the interval \(\mu±2\sigma\) is approximately 99.89%.

Step-by-step solution

Apply Poisson distribution formula

The Poisson distribution formula is: \(P(X=k)=\frac{e^{-\lambda}\lambda^{k}}{k!}\) Here, \(\lambda\) is the average rate (mean), \(e\) is the base of the natural logarithm (approximately 2.71828), and \(k\) is the number of events (neighbors in this case). The density of seedlings is 4 per square meter, so we have \(\lambda=4\) in this problem.

Determine the probability of no neighbors

We want to find the probability of zero neighbors (\(P(X=0)\)). Use the Poisson distribution formula with \(\lambda=4\) and \(k=0\): \(P(X=0)=\frac{e^{-4}\cdot4^{0}}{0!}\approx0.0183\) The probability that there are no neighbors within a square meter is roughly 1.83%.

Determine the probability of at most 3 neighbors

We want to find the probability for up to 3 neighbors (\(P(X\leq3)\)). We'll compute the probabilities for \(k=0,1,2,3\) and add them: \(P(X\leq3)=P(X=0)+P(X=1)+P(X=2)+P(X=3)\) \(=\frac{e^{-4}\cdot4^0}{0!}+\frac{e^{-4}\cdot4^1}{1!}+\frac{e^{-4}\cdot4^2}{2!}+\frac{e^{-4}\cdot4^3}{3!}\approx0.4335\) The probability of having at most 3 neighbors within a square meter is roughly 43.35%.

Determine the probability of 5 or more neighbors

We want to find the probability for 5 or more neighbors (\(P(X\geq5)\)). Notice that \(P(X\geq5)=1-P(X\leq4)\). We'll compute the probabilities for \(k=0,1,2,3,4\) and subtract the sum from 1: \(P(X\geq5)=1-\left(P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)\right)\) \(=1-\left(\frac{e^{-4}\cdot4^0}{0!}+\frac{e^{-4}\cdot4^1}{1!}+\frac{e^{-4}\cdot4^2}{2!}+\frac{e^{-4}\cdot4^3}{3!}+\frac{e^{-4}\cdot4^4}{4!}\right)\approx0.3712\) The probability of having 5 or more neighbors within a square meter is roughly 37.12%.

Find the proportion in the interval \(\mu±2\sigma\)

Since \(\lambda\) is both the mean and variance of a Poisson random variable, we have \(\mu=\lambda=4\) and \(\sigma=\sqrt{\lambda}=\sqrt{4}=2\). Then, the interval is \(4\pm2(2)=[0,8]\). Computing the probability of having neighbors in the range \(0\) to \(8\): \(P(0\leq X\leq8)=\sum_{k=0}^{8}\frac{e^{-4}\cdot4^k}{k!}\approx0.9989\) Thus, approximately 99.89% of the time, a seedling would have neighbors within the interval \(4\pm2\sigma\). This indicates that the range of neighbors is quite small with a high probability, as expected from a Poisson distribution with a relatively low mean.

Key concepts

Probability

Probability is the measure of the likelihood that a specific event will occur. It ranges from 0 to 1, where 0 indicates impossibility, and 1 indicates certainty.

In the context of this problem, probability helps us understand and quantify how often a seedling might have a certain number of neighbors. The Poisson distribution is used here because it models the occurrence of events (neighbors) within a given fixed time or space, assuming these events happen independently.

• To calculate the probability of no neighbors, the formula \(P(X=k)=\frac{e^{-\lambda}\lambda^{k}}{k!}\) is used.
• Similarly, the probability for up to three neighbors is calculated by summing the probabilities for zero, one, two, and three neighbors.
• For calculating the probability of five or more neighbors, the sum of probabilities for having four or fewer neighbors is subtracted from one.

Understanding these calculations is important for estimating how crowded or isolated a seedling is likely to be in its resource zone.

Random Variable

A random variable is a variable whose value depends on the outcomes of a random phenomenon. In probability theory, it is used to quantify the outcomes in numerical terms.

Here, our random variable is the number of neighbors a seedling might have. It is a discrete variable, as it only takes on non-negative integer values (like 0, 1, 2, etc.).

• We denote this variable by \(X\), representing the number of neighbors.
• It follows a Poisson distribution because it deals with counting occurrences of neighbors over a constant area with an average rate.

This modeling allows us to make statements such as "what is the probability \(X\) equals a particular value," which translates into meaningful insights about plant interaction in an ecological model.

Mean and Variance

In statistical terms, the mean and variance are crucial descriptors of a probability distribution.

The mean is the average value, and the variance measures how much the values differ from the mean.

In a Poisson distribution:

• The mean \(\mu\) is also known as the expected value, representing the average number of occurrences.
• For our seedlings, the mean is 4 per \(\text{m}^2\), which is why \(\lambda=4\).
• The variance \(\sigma^2\) is equal to \(\lambda\) in a Poisson distribution, reflecting consistency in the spread of data points around the mean.

This equality between mean and variance tells us that while the average number of neighbors is 4, there's potential variability in this number, though part of this intervals' coverage in probability calculations.

Mathematical Model

A mathematical model is an abstract representation that uses mathematical language to describe the behavior of a system.

For this problem, the Poisson distribution serves as our mathematical model to predict and analyze resource competition among plant seedlings. This model helps in understanding complex biological interactions in a simplified way.

• It assumes random distribution of seedlings. This captures unpredictability in natural dispersal mechanisms like wind or animal activity.
• The model's parameter, \(\lambda\), is determined by the average density of seedlings.
• It simplifies complex ecological dynamics into a manageable quantitative form, allowing for straightforward probability calculations.

By using this model, ecologists and researchers can estimate how resource depletion zones might overlap, affecting growth and survival, simply from known average densities.