**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.

Given a system of equations dHdt=aH-cHLdLdt=-bL+dHL where a,b,c,d are positive constants.

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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