Project bi-weekly report - Week 2
Team 4: Khalil Kafieh (20344087), Enoch Jayakody (20320974), Katrina Reimer
(20299962)
Topic: Plants!
Part 1: Existing System of ODEs
The Lotka-Volterra competition model describes the interactions of two plant species.
𝑑𝑃1
𝑃1 + 𝛼𝑃2
= 𝑟1 𝑃1 (1 −
)
𝑑𝑡
𝐾1
𝑑𝑃2
𝑃2 + 𝛽𝑃1
= 𝑟2 𝑃2 (1 −
)
𝑑𝑡
𝐾2
The population sizes of each species are denoted by P1 and P2. Intrinsic growth rates of each are
represented by r. K represents the maximum population size for the given ecosystem. Alpha denotes
the effects P2 has on P1, and beta is the effect P1 has on P2. This is a simplified version of the model,
which allows for differing effects each population has on the other.
This is a system first order nonlinear ODEs. It is a nonhomogeneous system because each equation
involves both P1 and P2.
The purpose of this model is to show how two plant species coexist, one as the predator and one as
the prey.
Part 2: Existing System of ODEs
Sure, let's consider a second-order ordinary differential equation (ODE) that models the growth of a
population of plants. The equation can represent the change in population over time, based on
factors such as reproduction and competition for resources. Let's represent the population of plants
as 𝑃(𝑡), where t is time.
The differential equation is:
𝒅𝟐 𝑷
𝒅𝑷
+𝒂
+ 𝒃𝑷 = 𝟎
𝟐
𝒅𝒕
𝒅𝒕
𝒅𝟐 𝑷
𝒅𝑷
Represents the acceleration of population growth, 𝒅𝒕 Represents the rate of change of
𝒅𝒕𝟐
population, a is a constant that captures the effect of factors influencing the rate of change and b is
a constant that represents the intrinsic growth rate of the population.
Now, solve this second-order ODE.
The characteristic equation for this ODE is obtained by assuming a solution of the form 𝑃(𝑡) =
e^(r1t), where r is a constant. Substituting this into the ODE, we get:
𝒓𝟐 + 𝒂𝒓 + 𝒃 = 𝟎
Solving this quadratic equation gives us the roots r1 and r2. The general solution for the ODE is then a
linear combination of the solutions associated with these roots: 𝑷(𝒕) =c1e^(r1t) + c2e^(r2t)
Here, c1 and c2 are constants determined by the initial conditions.
The specific values of r1 and r2 will determine the behaviour of the population over time. If r1 and r2
are real and negative, the population will approach an equilibrium over time. If they are complex
conjugates with a negative real part, the population will oscillate.
This example exhibits the utility of second-order ODEs in modelling population dynamics and can be
adapted to represent various scenarios in plant ecology.