Problem 12 - Calculus for Biology and Medicine

Question

(Adapted from Crawley, 1997) Denote plant biomass by \(V\), and herbivore number by \(N .\) The plant-herbivore interaction is modeled as $$ \begin{array}{l} \frac{d V}{d t}=a V\left(1-\frac{V}{K}\right)-b V N \\ \frac{d N}{d t}=c V N-d N \end{array} $$ (a) Suppose the herbivore number is equal to \(0 .\) What differential equation describes the dynamics of the plant biomass? Can you explain the resulting equation? Determine the plant biomass equilibrium in the absence of herbivores. (b) Now assume that herbivores are present. Describe the effect of herbivores on plant biomass; that is, explain the term \(-b V N\) in the first equation. Describe the dynamics of the herbivoresthat is, how their population size increases and what contributes to decreases in their population size. (c) Determine the equilibria (1) by solving $$ \frac{d V}{d t}=0 \quad \text { and } \quad \frac{d N}{d t}=0 $$ and (2) graphically. Explain why this model implies that "plant abundance is determined solely by attributes of the herbivore," as stated in Crawley (1997).

Step-by-Step Solution

Answer

(a) Without herbivores, plant biomass reaches equilibrium at carrying capacity \(K\). (b) Herbivores decrease plant growth via \(-bVN\), while their own population grows with \(cVN\) and decreases with \(-dN\). (c) Equilibrium is achieved when \(V = \frac{d}{c}\) and \(a = bN\); plant abundance is controlled by herbivore traits.

Step 1: Analyze Herbivore-Free Plant Dynamics

When the herbivore number, \(N\), is equal to 0, the plant biomass differential equation simplifies to: \[ \frac{dV}{dt} = aV\left(1-\frac{V}{K}\right). \] This is a logistic growth model where \(a\) is the intrinsic growth rate and \(K\) is the carrying capacity of the plant. The equation describes how plant biomass \(V\) grows exponentially when \(V\) is small, and levels off as \(V\) approaches \(K\).

Step 2: Equilibrium Without Herbivores

To find the equilibrium point, set \(\frac{dV}{dt} = 0\), which gives: \[ aV\left(1-\frac{V}{K}\right) = 0. \] Solving this, we get two equilibria: \(V = 0\) and \(V = K\). Thus, in the absence of herbivores, the plant biomass equilibrium is \(V = K\), meaning the plant population stabilizes at the carrying capacity.

Step 3: Impact of Herbivores on Plant Biomass

When herbivores are present, the term \(-bVN\) in the plant biomass equation \(\frac{dV}{dt} = aV\left(1-\frac{V}{K}\right)-bVN\) represents the consumption of plants by herbivores, reducing the growth rate of plant biomass. As \(N\) (the number of herbivores) increases, the negative effect on \(V\) becomes stronger, potentially preventing \(V\) from reaching \(K\).

Step 4: Dynamics of Herbivores

The equation \(\frac{dN}{dt} = cVN - dN\) describes how herbivore population changes. The term \(cVN\) depicts the growth in herbivore numbers due to feeding on plants, while \(-dN\) represents the natural mortality or other decrease factors in the herbivore population, such as competition or predation.

Step 5: Determine Equilibria Analytically

Equilibria occur when \(\frac{dV}{dt} = 0\) and \(\frac{dN}{dt} = 0\). Setting the first equation to zero: \[ aV\left(1-\frac{V}{K}\right) = bVN \Rightarrow V(a-bN) = \frac{aV^2}{K}. \]Setting the second equation to zero:\[ cVN = dN \Rightarrow V = \frac{d}{c}. \]Solve these simultaneously for \(V\) and \(N\) to find non-trivial equilibria.

Step 6: Interpret Equilibria Graphically

Graphically, the nullclines are lines where \(\frac{dV}{dt} = 0\) or \(\frac{dN}{dt} = 0\). The intersection points of these lines give equilibrium points. According to Crawley (1997), plant abundance is determined by herbivore attributes because, in the equilibrium where herbivores exist, the plant equilibrium \(V^*\) is solved from the herbivore equation \(V = \frac{d}{c}\), indicating dependency on herbivore traits.

Key Concepts

Differential Equations

Differential equations are fundamental tools in ecological modeling, particularly to describe and understand dynamics of biological systems like plant-herbivore interactions. These equations allow us to model the change of variables over time.

In the provided exercise, two differential equations are used: one for the plant biomass and another for the herbivore population size. The first differential equation: \[ \frac{dV}{dt} = aV \left(1 - \frac{V}{K} \right) - bVN \] depicts the dynamics of plant biomass. The term \(aV\left(1-\frac{V}{K}\right)\) models logistic growth, where \(a\) represents the plant's intrinsic growth rate and \(K\) its carrying capacity. The operation \(-bVN\) in the equation represents the pressure herbivores exert on plant biomass by consumption.

The second equation,\[ \frac{dN}{dt} = cVN - dN \]describes herbivore population dynamics. The term \(cVN\) indicates herbivore proliferation due to plant consumption, while \(-dN\) captures reductions due to mortality or loss factors. Understanding these equations helps decipher how plant and herbivore populations interact and affect each other's growth and sustainability.

Plant-Herbivore Interactions

Understanding plant-herbivore interactions is crucial for ecological studies as it helps to understand how these two populations influence each other. In this context, plant-herbivore interactions can either stabilize or destabilize ecosystems.

The main effect herbivores have on plants is captured by the \(-bVN\) term in the plant biomass differential equation. Here’s what this means:

• \(bVN\): Herbivore-driven consumption reduces plant biomass.
• Higher \(N\): Leads to greater reduction in \(V\), influencing the plant's ability to reach its carrying capacity (\(K\)).

The presence of herbivores means that plant growth isn't only limited by environmental resources but also by the number of herbivores. This interaction often results in plants not reaching their full biomass potential, hence altering the ecosystem balance.

While herbivores suppress plant population, they rely on those plants for food. This dependency creates a natural check-and-balance system crucial for maintaining biodiversity and ecosystem health.

Population Dynamics

Population dynamics study the changes in population sizes and can be influenced by a multitude of factors, which this model addresses. The differential equations presented are simplified models capturing the essence of how populations of plants and herbivores change over time.

### Key Concepts:

• Logistic Growth (without herbivores): Plants grow until they reach their environmental limitations (\(K\)).
• Impact of Herbivores: Herbivores consume plants, preventing populations from hitting \(K\).

Further examining the exercise, population dynamics are evident when solving for equilibria:

• Plant equilibrium (\(V\)) without herbivores: \(V = K\) (at carrying capacity).
• Equilibrium with herbivores: Influences from herbivore attributes determine plant abundance, demonstrating the interconnectedness of these systems.

Through both analytical and graphical solutions, it becomes clear that population dynamics in these models showcase the feedback loops and dependencies between different species. This gives insight into ecological balance, crucial for conservation efforts and understanding ecological resilience.