# An Enhanced Genetic Algorithm for Parameter Estimation of Sinusoidal Signals
## 2 Mathematical Ecological Theory Foundation  
In ecology, the number of offspring restricted to be equal to or less than that of parents is not common. Even though this may happen for reasons such as disease, food, water, and other factors, the species may become extinct or be restricted due to genetic competition as well. When the number of offspring is more than two, the population size grows rapidly with more competitive individuals [24], resulting in an increased chance for producing better offspring. Thus, it is desirable to have a larger number of individuals in genetic algorithms.  
Population dynamics were first studied by [PERSON] [36] in 1838, who describes the size and age composition of populations as dynamical systems driven by biological and environmental processes. The population dynamics model of a species can be mathematically expressed by:  
\[\frac{dP}{dt}=rP\cdot(1-\frac{P}{K}), \tag{1}\]  
where \(P\) represents population size, \(t\) is time, \(r\) defines the growth rate, and \(K\) is a constant denoting the environment carrying capacity. The solution to Equation (1) with an initial population \(P_{0}\) can be written as:  
\[P(t)=\frac{KP_{0}\cdot e^{rt}}{K+P_{0}\cdot(e^{rt}-1)}. \tag{2}\]  
Taking the birth and mortality rate into account, Equation (2) can be rewritten as:  
\[\Delta P(t)=\frac{KP_{0}\cdot e^{(r_{1}-r_{2})t}}{K+P_{0}\cdot[e^{(r_{1}-r_{2}) t}-1]}, \tag{3}\]  
where \(\Delta P\) is the total population size, and \(r_{1}\) and \(r_{2}\) are birth rate and mortality rate, respectively. Over time, the population dynamics presents different consequences in accordance with the relations between \(r_{1}\) and \(r_{2}\).  
1. When the birth rate is higher than the mortality rate, \(r_{1}>r_{2}\), the final population size for an infinite amount of time, \(t\), can be simplified as: \[\lim_{t\to\infty}\Delta P(t)=\lim_{t\to\infty}\frac{KP_{0}\cdot e^{(r_{1}-r_{2 })t}}{K+P_{0}\cdot[e^{(r_{1}-r_{2})t}-1]}=K\] (4) This means that a species can exist forever.
2. When the birth rate is equal to the mortality rate, \(r_{1}=r_{2}\), the exponential terms in the numerator and denominator become 1, so Equation (3) is given as follows: \[\lim_{t\to\infty}\Delta P(t)=P_{0}.\] (5) Here, as time approaches infinity, the final population size will equal to the initial value. What if the initial population size equals to 1 or 2? Although this species can live for a long time, the problem of self-reproduction and inbreeding results in an extinction.
3. When the birth rate is smaller than the mortality rate, \(r_{1}<r_{2}\), it is clear that \(e^{-x}\to 0\) as \(t\to\infty\). Hence, one can get the following expression: \[\lim_{t\to\infty}\Delta P(t)=\lim_{t\to\infty}\frac{KP_{0}\cdot e^{-(r_{2}-r_ {1})t}}{K+P_{0}\cdot[e^{-(r_{2}-r_{1})t}-1]}=0\] (6) Obviously, this equation represents an inevitable extinction of a species.  
According to the analysis above, on the one hand, it can be concluded that the extinction of a species occurs if the birth rate is equal to or less than the mortality rate; only when \(r_{1}>r_{2}\) can the species survive forever. On the other hand, it is found that the final population size strongly depends on both the growth rate and the time when the initial population size is fixed. Thus, in order to increase the number of offspring and get excellent individuals within a limited time, it is very necessary to improve the reproduction rate of a species [24]. In the case of the standard genetic algorithm (SGA), as the procreation is artificially controlled to be unchangeable, the extinction of the population of SGA is not allowed. Hence, neither is the evolutionary law consistent with the natural law, nor will it have a chance to acquire more excellent individuals. In this paper, based on the biology theory and mathematical ecological theory foundation, a genetic algorithm with several enhancements is proposed in order to achieve a higher accuracy, and faster convergence, and hence, to further improve the parameter estimation results of sinusoidal signals.