However, there are likely both lynx and hares around at the same time. We will model interactions between lynx and hare with a term using H - L in each equation. Our new system of equations is: dH = aH – cHL dt TP -bL + dHL dt where all of a, b, c, d are positive constants. Explain why, in terms of hares and lynx, there is a minus c and a plus d.
However, there are likely both lynx and hares around at the same time. We will model interactions between lynx and hare with a term using H - L in each equation. Our new system of equations is: dH = aH – cHL dt TP -bL + dHL dt where all of a, b, c, d are positive constants. Explain why, in terms of hares and lynx, there is a minus c and a plus d.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Mathematical Modeling of Predator-Prey Interactions**
We'll explore the interactions between hares and their primary predators, the lynx, using differential equations, similar to those in the S-I-R disease model. Here, \( H(t) \) represents the density of hares at time \( t \), while \( L(t) \) represents the density of lynx at time \( t \).
**Assumptions and Equations:**
1. **Hare Growth without Lynx:**
If lynx are absent, hares grow exponentially. This relationship is expressed by the differential equation:
\[
\frac{dH}{dt} = aH
\]
where \( a \) is a positive constant, indicating growth.
2. **Lynx Decline without Hares:**
In the absence of hares, lynx populations decline exponentially due to starvation. This is described by the equation:
\[
\frac{dL}{dt} = -bL
\]
where \( b \) is also a positive constant, representing the rate of decline.
These equations form the basis for understanding the dynamics between prey and predator populations under idealized assumptions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4e3da8d7-5e45-4b35-92f7-ecdb5bd91ee7%2Ff6fe4f2f-70b0-4f4f-a3c7-bb469776b3b1%2F2furypk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Mathematical Modeling of Predator-Prey Interactions**
We'll explore the interactions between hares and their primary predators, the lynx, using differential equations, similar to those in the S-I-R disease model. Here, \( H(t) \) represents the density of hares at time \( t \), while \( L(t) \) represents the density of lynx at time \( t \).
**Assumptions and Equations:**
1. **Hare Growth without Lynx:**
If lynx are absent, hares grow exponentially. This relationship is expressed by the differential equation:
\[
\frac{dH}{dt} = aH
\]
where \( a \) is a positive constant, indicating growth.
2. **Lynx Decline without Hares:**
In the absence of hares, lynx populations decline exponentially due to starvation. This is described by the equation:
\[
\frac{dL}{dt} = -bL
\]
where \( b \) is also a positive constant, representing the rate of decline.
These equations form the basis for understanding the dynamics between prey and predator populations under idealized assumptions.
![**Modeling Interactions Between Lynx and Hares**
In the study of ecological interactions, we often encounter situations where both lynx and hares coexist. To model these interactions mathematically, we incorporate a term using the product \(H \cdot L\) within each equation.
Our newly developed system of equations is presented as follows:
\[
\frac{dH}{dt} = aH - cHL
\]
\[
\frac{dL}{dt} = -bL + dHL
\]
In these equations, every variable and constant serves a purpose:
- \(H\) represents the hare population.
- \(L\) represents the lynx population.
- \(a, b, c, d\) are positive constants.
**Explanation of Terms:**
1. **Equation for \(\frac{dH}{dt}\)**: This equation describes the rate of change in the hare population.
- The term \(aH\) indicates the natural growth rate of hares.
- The term \(-cHL\) accounts for the decrease in the hare population due to predation by lynx.
2. **Equation for \(\frac{dL}{dt}\)**: This equation captures the rate of change in the lynx population.
- The term \(-bL\) reflects the natural mortality rate of lynx in the absence of hares.
- The term \(dHL\) signifies the growth contribution to the lynx population due to the availability of hares as prey.
The presence of a minus \(c\) in the hare equation and a plus \(d\) in the lynx equation highlights the predator-prey dynamics: hares decrease as they are preyed upon, whereas lynx benefit from the available prey, increasing their numbers.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4e3da8d7-5e45-4b35-92f7-ecdb5bd91ee7%2Ff6fe4f2f-70b0-4f4f-a3c7-bb469776b3b1%2F80lvmd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Modeling Interactions Between Lynx and Hares**
In the study of ecological interactions, we often encounter situations where both lynx and hares coexist. To model these interactions mathematically, we incorporate a term using the product \(H \cdot L\) within each equation.
Our newly developed system of equations is presented as follows:
\[
\frac{dH}{dt} = aH - cHL
\]
\[
\frac{dL}{dt} = -bL + dHL
\]
In these equations, every variable and constant serves a purpose:
- \(H\) represents the hare population.
- \(L\) represents the lynx population.
- \(a, b, c, d\) are positive constants.
**Explanation of Terms:**
1. **Equation for \(\frac{dH}{dt}\)**: This equation describes the rate of change in the hare population.
- The term \(aH\) indicates the natural growth rate of hares.
- The term \(-cHL\) accounts for the decrease in the hare population due to predation by lynx.
2. **Equation for \(\frac{dL}{dt}\)**: This equation captures the rate of change in the lynx population.
- The term \(-bL\) reflects the natural mortality rate of lynx in the absence of hares.
- The term \(dHL\) signifies the growth contribution to the lynx population due to the availability of hares as prey.
The presence of a minus \(c\) in the hare equation and a plus \(d\) in the lynx equation highlights the predator-prey dynamics: hares decrease as they are preyed upon, whereas lynx benefit from the available prey, increasing their numbers.
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