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Why can we not write the following predator prey model in matrix form X'(t) = AÑ(t)? x'(t) = ax(t) — ßx(t)y(t) y' (t) = xy(t) + 7x(t)y(t) -

Why can we not write the following predator prey model in matrix form X'(t) = AÑ(t)? x'(t) = ax(t) — ßx(t)y(t) y' (t) = xy(t) + 7x(t)y(t) -

Advanced Engineering Mathematics
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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### Predator-Prey Model and Matrix Representation #### Question: Why can we not write the following predator-prey model in matrix form \(\vec{X}'(t) = A\vec{X}(t)\)? #### Equations: 1. \(x'(t) = \alpha x(t) - \beta x(t) y(t)\) 2. \(y'(t) = -\lambda y(t) + \gamma x(t) y(t)\) #### Explanation: The system of differential equations above describes a predator-prey model where: - \(x(t)\) represents the prey population at time \(t\). - \(y(t)\) represents the predator population at time \(t\). - \(\alpha\), \(\beta\), \(\lambda\), and \(\gamma\) are constants that represent interaction rates. The key issue with writing this system in the form \(\vec{X}'(t) = A\vec{X}(t)\) is the presence of nonlinear terms like \(x(t)y(t)\). Matrix representation requires linearity, where derivatives are a linear combination of variables. Here, due to the products \(x(t)y(t)\) in the equations, the system exhibits nonlinearity, making it unsuitable for simple matrix representation.
Transcribed Image Text:### Predator-Prey Model and Matrix Representation #### Question: Why can we not write the following predator-prey model in matrix form \(\vec{X}'(t) = A\vec{X}(t)\)? #### Equations: 1. \(x'(t) = \alpha x(t) - \beta x(t) y(t)\) 2. \(y'(t) = -\lambda y(t) + \gamma x(t) y(t)\) #### Explanation: The system of differential equations above describes a predator-prey model where: - \(x(t)\) represents the prey population at time \(t\). - \(y(t)\) represents the predator population at time \(t\). - \(\alpha\), \(\beta\), \(\lambda\), and \(\gamma\) are constants that represent interaction rates. The key issue with writing this system in the form \(\vec{X}'(t) = A\vec{X}(t)\) is the presence of nonlinear terms like \(x(t)y(t)\). Matrix representation requires linearity, where derivatives are a linear combination of variables. Here, due to the products \(x(t)y(t)\) in the equations, the system exhibits nonlinearity, making it unsuitable for simple matrix representation.
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