Question

The giant lobelia Lobelia deckenii keniensis on Mount Kenya produces on average 250,000 seeds and flowers every eight years. Average adult survival is 0.984 per year. Plants do not begin setting seed until they are 50 years old. Assuming for simplicity a twostage life cycle (seeds, adult plants), calculate what survival rate of seeds would produce a stable population \((\lambda=1.0) .\) How would this survival rate change if the plants flowered every year instead of only once every eight years?

Short answer

Stable rate for the seeds over 8 years leads to \( s \approx 0.0000187 \). Annually, \( s \approx 0.0000635 \).

Step-by-step solution

Define the Model

We are dealing with a two-stage life cycle: seeds and adult plants. A stable population means that the population growth rate \( \lambda = 1.0 \). Therefore, the number of seeds produced must equal the number of seeds needed to maintain the stable adult population size.

Calculate Adult Plant Survival Over Eight Years

Given that the adult survival rate is 0.984 per year, to remain alive over the 8-year cycle between flowering events, we compute this as \( 0.984^8 \).

Compute Effective Number of Seeds per 8-Year Cycle

If an adult plant produces 250,000 seeds every 8 years, and the survival rate of adult plants over that period is 0.984 to the 8th power, then the effective number of seeds produced by each adult plant is \( 250,000 \times 0.984^8 \).

Set Up Equation for Seed Survival

To maintain a stable population, these produced seeds must survive to adulthood, meaning we set up the equation: Number of new adult plants = Effective number of seeds \( \times s \), where \( s \) is the seed survival rate. Since \( \lambda = 1.0 \), this equals 1 adult plant. Solve for \( s \).

Recalculate for Annual Flowering

If the plants flower every year, each adult plant would produce 250,000 seeds annually. Assuming an unchanged adult survival rate per year, we now calculate the new seed survival rate needed to maintain the stable size. The new equation would now reflect annual seed production.

Key concepts

Species Life Cycle

The life cycle of a species describes the sequence of stages that an organism goes through from birth to reproduction and eventually to death. In the case of the giant lobelia Lobelia deckenii keniensis, we consider a simplified two-stage life cycle consisting of seeds and adult plants. This simplification helps in understanding and modeling the dynamics of population growth and stability.
The giant lobelia spends many years in its life cycle:

• Seeds: The potential plant begins its life as a seed. This is a dormant stage until conditions are right for growth.
• Adult Plant: After surviving various conditions and stages, it becomes an adult, but it does not produce flowers until it reaches 50 years of age.

Understanding these stages is crucial because they influence the overall survival and reproductive strategy of the species, thereby affecting its population dynamics.

Survival Rate

Survival rate refers to the proportion of individuals surviving from one stage of life to another. In population biology, it's essential to track how well different stages survive as it impacts the overall growth of the population. In the case of adult lobelias, the average survival rate is 0.984 per year.
For a stable population, each adult lobelia must effectively replace itself after contributing to the next generation through seeds. The survival rate thus becomes a critical parameter in determining whether the population is growing, shrinking, or stable.
For lobelias:

• Seed Survival: It is crucial to determine the rate at which seeds survive to maturity through the calculation provided in the exercise.
• Adult Survival: This annual rate must also be considered over the flowering cycle to determine cumulative survival.

By calculating the survival rates at each life stage correctly, one can predict and influence the population outcome effectively.

Mathematical Modeling

Mathematical modeling involves using mathematical formulas and computations to simulate real-world processes. It's a way of using math to predict how populations will behave over time based on current data. This exercise applies mathematical modeling to predict the stable population of giant lobelias.
The process starts by defining growth rates and survival rates to set up equations that describe the population dynamics. For the lobelia:

• Calculate survival over a specified time: Use \[0.984^8\] to determine adult survival over eight years.
• Determine seed production: Consider seeds produced in an 8-year cycle, influenced by survival rates.
• Set up the equation for equilibrium: Relate seeds produced and seeds needed to maintain the population size where \(\lambda = 1.0\).

This mathematical approach effectively simplifies complex biological processes into understandable and solvable problems.

Population Stability

Population stability occurs when a population's growth rate is balanced such that it remains constant over time, meaning each adult just replaces itself with an offspring that survives to adulthood. For the giant lobelia, stability requires a careful balance between the seeds surviving to become adults and the rates at which adults survive and reproduce.
To achieve this:

• Determine effective seed production needed: Using the survival calculations, estimate how many seeds need to survive to adulthood.
• Maintaining \(\lambda = 1.0\): Proper balancing of survival from seeds to adults ensures the population remains stable.
• Adjustments for new conditions: Consider how changes, like annual flowering, affect stability for ongoing population success.

Understanding and maintaining population stability is crucial for conserving species in environments where conditions are changing, thereby ensuring their long-term survival.