1. Modeling and simulation of systems
Slovak University of Technology
Faculty of Material Science and Technology in Trnava
Modeling and simulation of systems
Simulation of continuous systems

2. Simulation of continuous systems
In general, the system is described by the system of differential equations
The relation between outputs and inputs is expressed by continuous transmissions.
The transmission is registered as picture in Laplace´s or Laplace-Wagner´s transformation

3. Simulation of continuous systems
X1(t)
X2(t)
System
X2(p)
X1(p)
Transmission S(p) has the form:
The transmission is equipollent to equation:

4. Simulation of continuous systems
By usage of substitution is possible to decrease the rule of differential equation up to equation of the first rule.
The methods of numerical integration are used for solution.
Euler´s method
Method of Runge-Kutta
Method of prediktor-korektor

5. Mathematical model of biological system
Pray
Predator
Elements of biological system
pray - mark x
predator - mark y
The predator influences the population of pray by hunting.
The pray serves as food for predators

6. Mathematical model of biological system
Growth of pray on the interval t,t+t xn(t) = k1x(t). t k1 – relative natality of pray
The number of killed prays on the interval t,t+t xm(t) = k2x(t). y(t). t k2 – predation of predator
The whole change of the number of pray in time t x(t) = k1x(t) - k2x(t) y(t) . t

7. Mathematical model of biological system
The growth of predators on the interval t,t+t yn(t) = k3k2y(t)x(t). t k3 – natality of predators
Mortality of predators in t ym(t) = k4y(t). t k4 – coefficient of mortality
The whole change of number of predators in time t y(t) = k3k2y(t)x(t) - k4 y(t) . t

8. Mathematical model of biological system
Then the system will be described by mathematical model - by the system of differential equations of the first rule
It is necessary to fill in this entry by the number of prays and
predators in time t=0, then x(0), y(0).