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Analysis of Hungry Wolf Chasing Rabbits Based on Higher Order Ordinary Differential Equation Model
Zhu Yunlong 1, Zhao Na 2, Sun Lijie 1, Wang Bo 1, Cheng Ming 1, Bai Haitao 1,
Wang Jian 1, Li Kai 1, Zhao Fuxing 1, Wang Tiezhu 1.
1 Department of Mining Engineering, Liaoning Technical University, Fuxin, Liaoning (123000)
Department of Bioengineering (Food Science), Liaoning Technical University, Fuxin, Liaoning (123000)
E-mail: zyl275887234@ 163.com
Abstract: The high-order ordinary differential model is used to study whether the hungry wolf can catch up with the rabbit. Firstly, the trajectory models of wolves and rabbits are established.
The rabbit ran straight to the cave in the north, and the wolf followed the curve. Then, draw the luck of wolves and rabbits with matlab.
Moving trajectory diagram. Then the value of y when x=0 is obtained by analytical method, so as to judge whether the wolf can catch up with the rabbit in turn. most
After that, the numerical differential method is used to solve the value of y when x=0 to judge whether the wolf can catch the rabbit before it enters the hole. This is a delicious meal.
Dunn. Ordinary differential equations have important applications in many disciplines, such as automatic control, the design of various electronic devices, missiles and so on.
Channel calculation, research on flight stability of aircraft and missiles, research on stability of chemical reaction process, etc. These questions can be
In order to find the solution of ordinary differential equation.
Keywords: higher order ordinary differential; Numerical differentiation; mathematical model
China Library Classification Number: O 172 438+0
1 Introduction
In our real life, there are many problems to pursue, such as racing, track and field competitions, the eagle catching rabbits and so on.
So are these questions valid and successful? We will discuss and verify the models of wolves and rabbits again to see if we can
Catch up and draw the curve of wolf and rabbit by MATLAB [1]. There are many places where we can use this in our real life.
Some chase models. Although the wolf has no time to take care of the rabbit's cave and calculate how to catch up with the rabbit, all it loses is.
This is just a good meal, so we can find other prey. But we humans are different, for example, in the military, chasing China missiles.
The problem of enemy planes is similar to the model of hungry wolves chasing rabbits. According to the distance between the tracker and the tracked person and the escape situation of the tracked person
Range, through calculation, adjust the speed appropriately, you can catch up. If you pursue it without thinking, the consequences will be unimaginable and the loss will outweigh the gain.
It will be more than just a meal. Therefore, through the analysis of this model, a clear MATLAB curve will be obtained.
Let the results clearly appear on the computer, at a glance, I hope this model can be used in our real life and get some use.
Improve the application of national economy and science and technology.
Step 2 ask questions
In the mysterious nature, dangers are hidden everywhere. Hunting and escaping play a vital role in the survival of animals.
Running speed and route are the key factors to catch up and escape. What we're talking about here is chasing old enemies and running fast.
Can a wolf catch up with a rabbit in a cave not far away?
There is a rabbit and a wolf. The rabbit is located 100 meters west of the wolf. Suppose the rabbit and the wolf find each other at the same time and come together.
At first, the rabbit ran to the nest 60 meters north, while the wolf was chasing the rabbit. As we all know, rabbits and wolves run at a constant speed, while the speed of wolves is
Twice as many as rabbits. Try to establish a mathematical model [2] to study the following problems:
(1) According to the known conditions, the differential model of the wolf trajectory is established.
(2) Draw the trajectories of rabbits and wolves.
(3) Using analytical method to determine whether the rabbit can safely return to the nest.
(4) Using numerical method to judge whether rabbits can safely return to their nests.
3 model construction
Suppose the wolf doesn't know whether there is a cave in the distance of the rabbit, then the speed direction of the wolf should always be towards the rabbit, but the rabbit is not.
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Deviation, so the speed direction of the wolf is constantly changing, and the trajectory should be a smooth curve. Set the speed of the rabbit.
V, taking the position of the rabbit when t=0 as the origin, the direction of the rabbit towards the wolf as the X axis, and rotating 90 degrees counterclockwise.
Establish a plane rectangular coordinate system in the y-axis direction. At time t, the coordinates of the wolf are (x, y), the coordinates of the rabbit are (0, vt), and the speed of the wolf.
The angle between the degree direction and the negative semi-axis of X axis is θ.
3. 1 problem analysis and model building
3.3. 1 Establish the differential model of the wolf trajectory.
Draw a sketch of the wolf's trajectory as follows:
Figure 1 Schematic diagram of the wolf's trajectory
Figure 1 the trajectory of the wolf plan
The derivative of y to x at time t is equal to the tangent slope of the curve at point (x, y), i.e.
Y=？ tanθ ( 1)
And because the wolf's movement direction points to the rabbit, so,
x
vt？ y
tanθ = =？ tanθ
Advanced (short for deluxe)
dysprosium (Dy)
(2)
Derived from (1) and (2),
x
y vt
Advanced (short for deluxe)
dy？
=
(3)
The wolf's speed is decomposed into x-axis and y-axis directions, that is, x v =
Advanced (short for deluxe)
dt，
y
v dy
Trembling insanity (abbreviation for Delirium Tremens)
=
, so,
2
2 2
(2v)
Trembling insanity (abbreviation for Delirium Tremens)
Advanced (short for deluxe)
Trembling insanity (abbreviation for Delirium Tremens)
dy =
+ ?
(4)
Can be obtained from formula (3),
y = x dx
dysprosium (Dy)
+ vt (5)
Take the derivative of t on both sides,
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v
Trembling insanity (abbreviation for Delirium Tremens)
Advanced (short for deluxe)
Advanced (short for deluxe)
x d y
Advanced (short for deluxe)
dysprosium (Dy)
Trembling insanity (abbreviation for Delirium Tremens)
Advanced (short for deluxe)
Advanced (short for deluxe)
dy =？ + ? + 2
2
(6)
Pack up, take it
Trembling insanity (abbreviation for Delirium Tremens)
Advanced (short for deluxe)
Advanced (short for deluxe)
x d y？ 2
2
= ? Five (7)
Multiply the left and right sides of Formula (4) by the same value.
2 dt
Advanced (short for deluxe)
Have to
2 years
Advanced (short for deluxe)
+ 1=
2
2 4
Advanced (short for deluxe)
V. dt (8)
Derived from (7) and (8)
2
2
Advanced (short for deluxe)
Dee
v
x
Advanced (short for deluxe)
dt =？
(9)
Equation (9) is a differential model of the trajectory of the wolf.
3.3.2 Draw the trajectories of rabbits and wolves.
According to the above differential equation, X in the second-order differential equation (9) can be obtained by using the ode45 function in matlab software.
Value corresponding to the y value, and then use the drawing function plot to draw the wolf's trajectory image [3]. The process is as follows:
Firstly, the matlab function is established:
Function f=odefun(x, y)
f( 1， 1)= y(2)；
f(2， 1)=sqrt( 1+y(2)。 ^2)./(2.* x)；
Then enter the following program in the main program:
t = 100:-0. 1:0. 1；
y0 =[0 0]；
[T，Y] = ode45('odefun '，T，y0)；
plot(T，Y(:， 1)，'-')
You can get the following curve, which is the track map of the wolf.
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Fig. 2 the track of the wolf.
Fig. 2 the trajectory of the wolf.
The rabbit's trajectory is a straight line from (0,0) to its cave (0,60), so in the main program,
Enter the following procedure and draw the tracks of rabbits and wolves.
x 1 =[0 0]；
y 1 =[0 60]；
plot(T，Y(:， 1)，'-'，x 1，y 1，' r ')
The drawn image is as follows:
(where blue represents the path of the wolf and red represents the path of the rabbit)
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Fig. 3 The movement track of wolves and rabbits.
Fig. 3 The trajectory of wolves and rabbits.
4 model solving
4. 1 Use analytical method to find out whether rabbits can safely return to their nests.
To judge whether the wolf can catch up with the rabbit, we can first assume that there is no cave and see where the wolf can catch up with the rabbit. If it catches up,
When the rabbit moved more than 60 meters, that is to say, before the wolf caught up with the rabbit, the rabbit had fled back to the cave safely.
In the middle. The specific process of judging whether the wolf can catch up with the rabbit by analytical method [4] is as follows:
assumptive
p dx
dysprosium (Dy)
=, then
2
2
dp d y
dx dx
=, then equation (9) can be changed to
2
2 2 4 1
+ = ?
Advanced (short for deluxe)
data processing
v
p v x ( 10)
arrange
2
2 2 4 1
+ = ?
Advanced (short for deluxe)
p v dp ( 1 1)
Advanced (short for deluxe)
p2 + 1 = 2x dp ( 12)
x
Advanced (short for deluxe)
p
data processing
2 1 2
=
+
( 13)
Then combine the two sides of the equation, and you get
( ) '
1 ln p+p2+ 1 = ln x+C( 14)
namely
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p+p2+ 1 = C x 1( 15)
Because when x= 100, the speed direction of the wolf is negative along the y axis, so at this time p=0, we can get 1 C =
1
10
Equation (15) can become
p + p2 + 1 = x
10
1
( 16)
Square on both sides
100
2p 2+ 1+2p p2+ 1 = x( 17)
Interchange of terms
2 p p2 + 1 = (2 1)
100
x？ p2 +
( 18)
It's the second time
(2 1)
100
4 4 1 2
10000
4 4 4 2 2
2
p4 + p2 = x + p + p +？ x p + ( 19)
arrange
( ) 1 0
100
4 2
10000
2
2
x？ p + x + =
(20)
Seek p
2
2
2 10
10
100 2
100
2
100
1
4 10000
= + ? = ?
+
=
x
x
x
x
x
x
p
(2 1)
x
p x 5
20
= ? (22)
because
p dx
dysprosium (Dy)
=, so equation (22) can be changed to
x
x
Advanced (short for deluxe)
dy 5
20
= ? (23)
The functional relationship between y and x can be obtained by integrating on both sides.
3 1
2 2
2
1 10
30
y = x？ x +C (24)
Because when x= 100, y=0, so
3 1
2 2
2
0 1 100 10 100
30
= ? +C
solve
2 degrees Celsius =
200
three
=66.67
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Therefore, equation (24) can become
3 1
1 2 10 2 200
30 3
y = x？ x + (25)
Let x=0, and you can get y=
200
three
=66.67
Because y = 66.67 >；; 60, so before the wolf catches up with the rabbit, the rabbit has fled back to the cave safely, and the hungry wolf can only
In a daze
4.2 Using numerical method to solve whether rabbits can safely return to their nests.
It has been judged by analytical method that the wolf did not catch up with the rabbit, and now it can be solved by numerical differential method (9).
The value of y when x=0 in the formula, and then compare the value of y with 60. If y is greater than 60, it also means that before the rabbit escapes safely back to the cave,
The wolf failed to catch up with the rabbit. The following is a method to judge whether the wolf can catch up with the rabbit by numerical differentiation and matlab software.
Method:
The initial value of the second-order ordinary differential equation is obtained by using the ode45 function in matlab software, and the value of Y when x= 100 is obtained.
It can be judged whether the wolf can catch up with the rabbit [5]. The specific matlab program is as follows:
Firstly, the odefun function is established:
Function f=odefun(x, y)
f( 1， 1)= y(2)；
f(2， 1)=sqrt( 1+y(2)。 ^2)./(2.* x)；
Then enter the following program in the main program:
t = 100:-0. 1:0. 1；
y0 =[0 0]；
[T，Y] = ode45('odefun '，T，y0)；
n=size(Y， 1)；
Y(n， 1)
You can output the results:
ans =63.5007
When x=0. 1, y = 63.5007 & gt60, and when x=0, y > 0；; Of course, 63.5007 is greater than 60, so the wolf is before the rabbit goes into the hole.
Did not catch up with the rabbit, so a good meal disappeared from its eyes.
5 result analysis
From Figure 2, we can see that the value of y is greater than 60 when x=0, and the value of y is equal to 66.67 and greater than that calculated by mathematical analysis.
60, the y value calculated by numerical differentiation method is also greater than 60. So from various calculation methods, it shows that rabbits are like caves.
Before, wolves couldn't catch them.
Think about it from another angle. Suppose the wolf knows where the rabbit's cave is and runs directly to its cave to wait for the rabbit. Nagen
According to Pythagorean Theorem [6], the distance of wolf movement is S = 602+1002 =16.6m, and the distance of rabbit movement is S/2 = 58.3.
In other words, the rabbit has not escaped into the hole, and the wolf has been waiting at its hole, so the rabbit is afraid to enter the hole, as long as the rabbit does not.
When the method enters the hole, the speed of the wolf is twice that of the rabbit, and the wolf can capture it. Unfortunately, hungry and greedy wolves only think about how.
Hurry up and catch up with the rabbit and have a good meal. There is no time and complicated calculation, and in many cases it is a wolf.
I don't know where the rabbit's cave is, but the wolf can only watch the rabbit slip away and be in a daze when it is about to catch up.