Question

Throughout its lifetime, itself a random variable having distribution function \(F(x)\), an organism produces offspring according to a nonhomogenous Poisson process with intensity function \(\lambda(u) .\) Independently, each offspring follows the same probabilistic pattern, and thus a population evolves. Assuming $$ 1<\int_{0}^{\infty}\\{1-F(u)\\} \lambda(u) d u<\infty $$ show that the mean population size \(m(t)\) asymptotically grows exponentially at rate \(r>0\), where \(r\) uniquely solves $$ 1=\int_{0}^{\infty} e^{-r u}\\{1-F(u)\\} \lambda(u) d u $$

Short answer

By the provided conditions and application of the monotone convergence theorem, we can justify the existence of a unique \(r>0\) that satisfies the given equation. Consequently, the mean population \(m(t)\) asymptotically grows exponentially at rate \(r\).

Step-by-step solution

Understand the Problem Setting

The problem describes a specific population model where each organism produces offspring according to a nonhomogeneous Poisson process, with intensity function λ(u). The entire lifetime of an organism is denoted as a random variable following a distribution function F(x). Every offspring independently follows this same pattern, leading to an evolving population over time.

Understand the Conditions

It's given that: \(\int_{0}^{\infty} \{1-F(u)\} \lambda(u) d u\) is finite and greater than 1. This constraint is crucial. It ensures that at any given time, the expected number of new offspring produced by the population as a whole is less than 1 and finite. This makes it possible to establish an exponential growth rate of the mean population size.

Identify the Required Proofs

The goal is to show that the mean population size \(m(t)\) grows exponentially at rate \(r>0\). To do this, it must be proven that \(r\) uniquely solves the equation \(1 = \int_{0}^{\infty} e^{-r u}\{1-F(u)\} \lambda(u) d u\). This mathematical model allows for the calculation of an exponential growth rate, considering both the variability in individual lifetime and reproduction behavior.

Deduction of the growth rate

Since \(\int_{0}^{\infty}\{1-F(u)\} \lambda(u) d u\) is finite and greater than 1, by the monotone convergence theorem we can infer that there exists a unique \(r>0\) such that \(1 = \int_{0}^{\infty} e^{-r u}\{1-F(u)\} \lambda(u) d u\). Because the offspring have an identical distribution pattern with their parents, the population growth rate will stabilize to this same exponential rate \(r\). Therefore, the mean population size \(m(t)\) will grow exponentially at rate \(r\).

Key concepts

Nonhomogeneous Poisson Process

The nonhomogeneous Poisson process is an essential concept in modeling events that occur randomly over time. Unlike the homogeneous Poisson process, where events occur at a constant average rate, the nonhomogeneous process allows this rate, called the intensity or rate function, to change over time. This variability makes it particularly useful for biological models where the rate of occurrences, such as births or deaths, can depend on various factors like the age of an organism or environmental conditions.

In the context of the exercise, the offspring production rate is modeled using a nonhomogeneous Poisson process with a rate function \(\lambda(u)\). Understanding this concept is crucial as it lays the groundwork for predicting how the population size changes over time.

Random Variable

A random variable is a numerical outcome of a random phenomenon. In our exercise, the lifetime of organisms is treated as a random variable, and its unpredictability is captured mathematically by the distribution function \(F(x)\). The concept of a random variable is a cornerstone of probability and statistics because it enables us to quantify and analyze the randomness inherent in various processes, from quantum physics to finance to population biology.

By associating the organism’s lifespan with a random variable, we introduce a layer of real-world complexity to the population model. It acknowledges that every organism doesn’t live for a predetermined amount of time, rather it varies, often influenced by both genetic and environmental factors.

Exponential Growth

Exponential growth occurs when a quantity increases at a rate proportional to its current value, leading to a rapid escalation over time. It’s commonly expressed mathematically by the function \(y(t) = y_0e^{rt}\), where \(y_0\) is the initial quantity, \(r\) is the growth rate, and \(t\) is time. The phenomenon of exponential growth is omnipresent in nature, especially in populations where the rate of reproduction can lead to a snowballing effect on population size.

In our exercise, we're asked to demonstrate that the mean population size \(m(t)\) asymptotically grows at an exponential rate \(r>0\). The challenge lies in identifying this growth rate \(r\) which is deeply intertwined with the lifespan distribution and offspring production rate, emphasizing the interdependency of biological factors in growth scenarios.

Intensity Function

The intensity function, often denoted by \(\lambda(u)\), describes how the rate of a nonhomogeneous Poisson process varies with time or some other parameter, like age in the case of an organism's life cycle. It’s a vital element of the Poisson process which distinguishes it from the homogeneous counterpart, allowing for more sophisticated and realistic models of real-world phenomena.

In population dynamics, the intensity function is used to project the frequency of births or deaths within the population. As seen in the exercise, the offspring production follows a nonhomogeneous Poisson process where the intensity function \(\lambda(u)\) interacts with the lifespan distribution to shape the overall growth of the population.

Distribution Function

A distribution function, typically represented by \(F(x)\), characterizes the probability that a random variable takes a value less than or equal to \(x\). This function provides a complete description of the probability distribution of a random variable and is fundamental for calculating probabilities and expectations.

The importance of the distribution function in our exercise lies in capturing the probabilistic nature of organisms' lifespans. It is integrated with the intensity function in the integral equation to solve for the exponential growth rate \(r\), portraying how individual member’s probabilities accumulate to generate predictable patterns for the whole population.