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### Discrete Model for Population Dynamics The given population dynamics model is: \[ x_{n+1} = x_n \exp\left[r \left( 1 - \frac{x_n}{K} \right) \right] \] where: - \(x_n\) represents the population at time step \(n\), - \(r\) is the growth rate, - \(K\) is the carrying capacity. ### Steady States of the Model A steady state occurs when the population remains constant over time, meaning \(x_{n+1} = x_n = x^*\). Substituting \(x^*\) into the equation: \[ x^* = x^* \exp\left[r \left(1 - \frac{x^*}{K} \right)\right] \] This can be simplified to: \[ 1 = \exp\left[r \left( 1 - \frac{x^*}{K} \right)\right] \] Taking the natural logarithm of both sides: \[ 0 = r \left( 1 - \frac{x^*}{K} \right) \] From this equation, we find two possible steady states: 1. \( x^* = K \), which is the carrying capacity ___ ____ _______ _____ __________ _____ ______ ____ __________ ___ ______ ____.
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