Problem 2 - Calculus for Biology and Medicine

Question

Suppose the densities of two species evolve in accordance with the Lotka- Volterra model of interspecific competition. Assume that species 1 has intrinsic rate of growth \(r_{1}=4\) and carrying capacity \(K_{1}=17\) and that species 2 has intrinsic rate of growth \(r_{2}=1.5\) and carrying capacity \(K_{2}=32 .\) Furthermore, assume that 15 individuals of species 2 have the same effect on species 1 as 7 individuals of species 1 have on themselves and that 5 individuals of species 1 have the same effect on species 2 as 7 individuals of species 2 have on themselves. Find a system of differential equations that describes this situation.

Step-by-Step Solution

Answer

System of equations: \[ \frac{dN_1}{dt} = 4 N_1 \left( 1 - \frac{N_1 + \frac{7}{15}N_2}{17} \right) \] \[ \frac{dN_2}{dt} = 1.5 N_2 \left( 1 - \frac{N_2 + \frac{7}{5}N_1}{32} \right) \]

Step 1: Define the Lotka-Volterra Model for Interspecific Competition

The Lotka-Volterra equations for interspecific competition describe how two species' populations interact. The equations are given by:\[ \frac{dN_1}{dt} = r_1 N_1 \left( 1 - \frac{N_1 + \alpha_{12}N_2}{K_1} \right) \]and\[ \frac{dN_2}{dt} = r_2 N_2 \left( 1 - \frac{N_2 + \alpha_{21}N_1}{K_2} \right) \], where:- \( N_1 \) and \( N_2 \) are the population sizes of species 1 and 2, - \( r_1 \) and \( r_2 \) are the intrinsic growth rates,- \( K_1 \) and \( K_2 \) are the carrying capacities,- \( \alpha_{12} \) is the effect of species 2 on species 1,- \( \alpha_{21} \) is the effect of species 1 on species 2.

Step 2: Determine the Effect Coefficients \(\alpha_{12}\) and \(\alpha_{21}\)

Given that 15 individuals of species 2 have the same effect on species 1 as 7 individuals of species 1, we have:\[ \alpha_{12} = \frac{7}{15} \]Similarly, 5 individuals of species 1 have the same effect on species 2 as 7 individuals of species 2, we have:\[ \alpha_{21} = \frac{7}{5} \].

Step 3: Substitute Values Into the Model Equations

We substitute the given values into the Lotka-Volterra equations:For species 1:\[ \frac{dN_1}{dt} = 4 N_1 \left( 1 - \frac{N_1 + \frac{7}{15}N_2}{17} \right) \]For species 2:\[ \frac{dN_2}{dt} = 1.5 N_2 \left( 1 - \frac{N_2 + \frac{7}{5}N_1}{32} \right) \]

Key Concepts

Differential Equations

Differential equations are mathematical tools that describe how quantities change over time. They are crucial in modeling systems where change is expected. For two competing species, like in the Lotka-Volterra model, differential equations help us understand how each population evolves in response to various factors. In our example, the growth for each species is described by a differential equation:

• For species 1: \( \frac{dN_1}{dt} = 4 N_1 \left( 1 - \frac{N_1 + \alpha_{12}N_2}{17} \right) \)
• For species 2: \( \frac{dN_2}{dt} = 1.5 N_2 \left( 1 - \frac{N_2 + \alpha_{21}N_1}{32} \right) \)

Here, \( \frac{dN_1}{dt} \) and \( \frac{dN_2}{dt} \) represent the rate of change of population size over time. These equations involve parameters like the intrinsic growth rates \( r_1 \) and \( r_2 \), and carrying capacities \( K_1 \) and \( K_2 \). The interactions between the species through coefficients \( \alpha_{12} \) and \( \alpha_{21} \) also influence the outcome.

Interspecific Competition

Interspecific competition refers to the interaction between different species competing for the same resources in an environment. Such competition can affect population sizes, growth rates, and survival. In the Lotka-Volterra model used here, competition affects each species through competition coefficients:

• \( \alpha_{12} \) influences how species 2 affects the growth of species 1.
• \( \alpha_{21} \) measures the impact of species 1 on species 2.

For example, given that 15 individuals of species 2 impact species 1 the same way 7 individuals of species 1 do, the coefficient \( \alpha_{12} \) is calculated as \( \frac{7}{15} \). Similarly, \( \alpha_{21} = \frac{7}{5} \) reflects how 5 individuals of species 1 affect species 2 similarly to 7 of species 2.These coefficients help predict competition outcomes, by incorporating both interspecies and intraspecies interactions in the model.

Population Dynamics

Population dynamics is the study of how populations change over time due to births, deaths, immigration, and emigration. The Lotka-Volterra model provides a framework to analyze such dynamics in ecological settings with competing species. It is essential to:

• Identify factors that influence population growth, such as food availability and predation.
• Incorporate natural limits to growth using terms like carrying capacity, which restrict how large a population can become.

In this model, each species' population is affected by its inherent growth characteristics and interactions with other species. These factors combine to produce dynamic systems where population sizes can fluctuate unpredictably over time. By solving the differential equations, we can predict these changes and potentially uncover patterns in ecosystem behavior.

Carrying Capacity

Carrying capacity refers to the maximum population size an environment can sustain indefinitely without resource depletion. In the context of the Lotka-Volterra model, carrying capacities are vital for understanding limits to population growth.the equation for each species is adjusted for its respective carrying capacity:

• Species 1: \( K_1 = 17 \)
• Species 2: \( K_2 = 32 \)

These values cap the population size for each species in the absence of interspecies competition.Carrying capacity influences how populations stabilize and inform us about resource availability and habitat limitations. Different environmental conditions or resource distributions might alter these values, further emphasizing their importance in ecological models. Understanding carrying capacities helps us evaluate the sustainability of species and ecosystems over the long term.