Problem 13 - Calculus

Question

Humans vs Zombies \(^{29}\) is a game in which one player starts as a zombie and turns human players into zombies by tagging them. Zombies have to "eat" on a regular basis by tagging human players, or they die of starvation and are out of the game. The game is usually played over a period of about five days. If we let \(H\) represent the size of the human population and \(Z\) represent the size of the zombie population in the game, then, for constant parameters \(a, b,\) and \(c,\) we have: $$\begin{array}{l} \frac{d H}{d t}=a H Z \\ \frac{d Z}{d t}=b Z+c H Z \end{array}$$ (a) Decide whether each of the parameters \(a, b, c\) is positive or negative. (b) What is the relationship, if any, between \(a\) and \(c ?\)

Step-by-Step Solution

Answer

(a) \(a\) is negative, \(b\) is negative, and \(c\) is positive. (b) The relationship is usually \(c = -a\).

Step 1: Understanding Parameters

To begin understanding the behavior of the parameters \(a\), \(b\), and \(c\), we first identify the biological mechanics they might represent. \(a\) and \(c\) are coefficients in terms representing interactions between humans and zombies \((H \cdot Z)\), while \(b\) affects only the zombie population \((b \cdot Z)\).

Step 2: Analyzing the Human Population Equation

The equation for humans, \(\frac{dH}{dt} = aHZ\), shows that the rate of change in the human population depends on \(a\), the current number of humans \(H\), and zombies \(Z\). Since more interactions with zombies reduce human numbers (as they convert to zombies), \(a\) must typically be negative. This causes the decrease of humans when \(Z > 0\).

Step 3: Analyzing the Zombie Population Equation

The zombie equation \(\frac{dZ}{dt} = bZ + cHZ\) involves two terms: the first, \(bZ\), represents changes to zombie population independent of humans, such as natural death, indicating \(b\) is typically negative. The second term, \(cHZ\), represents conversion from humans to zombies due to interaction, suggesting \(c\) must be positive to reflect the increasing number of zombies.

Step 4: Relating Parameters \(a\) and \(c\)

The relationship between \(a\) and \(c\) is integral as both terms appear due to interactions between humans and zombies. Logically, \(c\) is the conversion factor making humans zombies, while \(-a\) is the factor diminishing humans. Hence, generally, \(c = -a\) if each human tagged becomes a zombie, since these actions are opposite processes of the same interaction.

Key Concepts

Population Dynamics

Population dynamics deals with how the population numbers of species change over time and what factors influence those changes. In the context of our game scenario, we have two populations to consider: humans and zombies. The size of these populations can fluctuate based on several conditions and interactions within the game.
In population dynamics:

• The human population decreases when a zombie tags a human, turning them into a zombie.
• The zombie population increases as more humans are converted but may also decrease if zombies fail to "eat" regularly, leading to their demise.

In this game, the challenge lies in the balance of these populations.
Understanding how each parameter affects growth or decline is crucial for predicting future population sizes.

Mathematical Modeling

Mathematical modeling involves creating mathematical formulations to approximate real-world situations. Here, we apply mathematical modeling to represent the interactions in the Humans vs Zombies game using differential equations.
The main idea of mathematical modeling in this scenario is to capture the essential dynamics of the game:

• The equation for humans, \(\frac{dH}{dt} = aHZ\), describes how humans are converted into zombies upon interactions, efficiently modelling this aspect of the game.
• The zombie equation, \(\frac{dZ}{dt} = bZ + cHZ\), not only considers how zombies grow by converting humans but also factors their natural decline due to starvation.

These equations use parameters like \(a\), \(b\), and \(c\) to encapsulate rates of interaction and population changes. By experimenting with different parameter values, one can understand diverse scenarios which can occur in the game.

Systems of Equations

Systems of equations are a set of equations with multiple variables that are solved together. In this problem, we have a system made up of two differential equations, each with variables \(H\) (humans) and \(Z\) (zombies).
Key points about systems of equations here:

• Each equation provides a unique perspective on population dynamics—one focuses on humans while the other focuses on zombies.
• Solving this system can tell us the outcome of the game, such as whether humans or zombies will dominate over time.

One main challenge in analyzing systems of equations is understanding the interplay between the variables across equations. In this case, it involves how effectively humans and zombies interact and the long-term sustainability of their populations.
By accurately analyzing these equations, players can strategize better around managing resources and movements within the game.