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Applications of differential equations 1. Sohag Em il : sohag. For Example, 5. In general , modeling variations of a physical quantity, such as temperature, pressure, displacement, velocity, stress, strain, or concentration of a pollutant, with the change of time t or location, such as the coordinates x, y, z , or both would require differential equations. Similarly, studying the variation of a physical quantity on other physical quantities would lead to differential equations.
Current at time t, i t , is the solution of the differential equation. LAW: The rate of change of the temperature of an object is proportional to the difference between its own temperature and the temperature of its surroundings.
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Example: 1. Example: 2. Example: 4. Coefficient depends on y Power not 1. Derivative form: 2. Differential form:. As is solution of the differential equation for every value of c1, hence it is known as general solution. Examples Observe that the set of solutions to the above 1st order equation has 1 parameter, while the solutions to the above 2nd order equation depend on two parameters.
The solution is a family of ellipses. For the family of straight lines the differential equation is Differential Equation Chapter 1 2. For the family of curves. Free falling stone where s is distance or height and g is acceleration due to gravity.
Spring vertical displacement Differential Equation Chapter 1 Differential Equation Chapter 1 where y is displacement, m is mass and k is spring constant 3. The inverse of the von Bertalanffy growth rate appears to depend linearly on the ultimate length, when different food levels are compared. The intercept relates to the maintenance costs, the slope to the rate at which reserve is mobilized for use by metabolism. The ultimate length equals the maximum length at high food availabilities. The theory attempted to provide alternatives to conventional models of organization.
Lotka in In Lotka extended, the model to "organic systems" using a plant species and a herbivorous animal species as an example and in he utilised the equations to analyse predator-prey interactions in his book on biomathematics arriving at the equations that we know today. Vito Volterra, who made a statistical analysis of fish catches in the Adriatic independently investigated the equations in The prey population finds ample food at all times.
Ordinary Differential Equations
The food supply of the predator population depends entirely on the prey populations. The rate of change of population is proportional to its size. During the process, the environment does not change in favour of one species and the genetic adaptation is sufficiently slow. As differential equations are used, the solution is deterministic and continuous.
This, in turn, implies that the generations of both the predator and prey are continually overlapping. If either x or y is zero then there can be no predation. With these two terms the equation above can be interpreted as: the change in the prey's numbers is given by its own growth minus the rate at which it is preyed upon.
Note the similarity to the predation rate; however, a different constant is used as the rate at which the predator population grows is not necessarily equal to the rate at which it consumes the prey. Hence the equation expresses the change in the predator population as growth fueled by the food supply, minus natural death.
If the initial conditions are 80 baboons and 40 cheetahs, one can plot the progression of the two species over time. The choice of time interval is arbitrary. As the predator population is low the prey population will increase again. These dynamics continue in a cycle of growth and decline. Population equilibrium Population equilibrium occurs in the model when neither of the population levels is changing, i.
Applications of Difference Equations in Biology |authorSTREAM
When solved for x and y the above system of equations yields and Hence, there are two equilibriums. The first solution effectively represents the extinction of both species. If both populations are at 0, then they will continue to be so indefinitely. The second solution represents a fixed point at which both populations sustain their current, non-zero numbers, and, in the simplified model, do so indefinitely. The replicator equation differs from other equations used to model replication, in that it allows the fitness landscape to incorporate the distribution of the population types rather than setting the fitness of a particular type constant.
This important property allows the replicator equation to capture the essence of selection.
The replicator equation does not incorporate mutation and so is not able to innovate new types or pure strategies. Since the elements of the population vector x sum to unity by definition, the equation is defined on the n-dimensional simplex. The replicator equation assumes a uniform population distribution; that is, it does not incorporate population structure into the fitness. The fitness landscape does incorporate the population distribution of types, in contrast to other similar equations, such as the quasispecies equation.
The analysis is more difficult and computationally intensive in the discrete formulation, so the continuous form is often used, although there are significant properties that are lost due to this smoothing. Note that the continuous form can be obtained from the discrete form by a limiting process.
- Applications of Difference Equations in Biology |authorSTREAM.
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The transformation is made by the change of variables where is the Lotka—Volterra variable.