Title: In the normal war model, it is assumed that the ratio of effective operational coefficients between Party A and Party B is a/b=4, and the initial forces x0 and y0 are the same.

(1) Ask Party B what the remaining strength is when winning, and how to determine the time for Party B to win.

(2) If Party A has reserve troops to reinforce at a constant speed R after the battle starts, re-establish the model and discuss how to judge the outcome of both sides.

Solution: In order to solve the above problems, we must build a model of conventional warfare. According to the requirements of the topic, based on the model in Section 5.3, we now establish the model as follows:

X (t) and y(t) are used to indicate the strength of both sides at time T, which can be regarded as the number of soldiers on both sides.

(1) The combat loss rate of both sides depends on the strength and combat effectiveness of both sides. The combat loss rates of Party A and Party B are expressed by f(x, y) and g (x, y) respectively.

(2) The non-combat attrition rate of both sides (caused by diseases, flying and other factors) is only proportional to their own strength.

(3) The reinforcement ratio of Party A and Party B is a given function, which is expressed by u(t) and v(t) respectively.

From this, we can write the differential equations about x (t) and y (t) as follows.

Equation (1)

When both Party A and Party B are fighting regular troops, we only need to analyze Party A's combat attrition rate f(x, y). J The public activities of Party A's soldiers are within the monitoring and killing range of every soldier of Party B. Once a soldier of Party A is killed, Party B's firepower will immediately focus on the rest of the soldiers, so the combat attrition rate of Party A is only related to Party B's strength, and it can be simply set that F is proportional to Y, that is, f=ay. A stands for the average killing rate of each soldier of Party B to the soldiers of Party A (the number of killings per unit time), which is called Party B's combat effectiveness coefficient ... A can be further decomposed into a=rypy, where ry is the firing rate of Party B (the number of shots per soldier per unit time) and py is the firing rate of each shot.

Similarly, g=bx, Party A's combat effectiveness coefficient B = Rpx, rx and px are Party A's firing rate and hit rate. Moreover, when analyzing the outcome of the war, we ignore the non-combat attrition (which is very small compared with the combat attrition), assuming that both sides have no reinforcements, and remembering that the initial forces of both sides are x0 and y0 respectively, the equation (1) can be simplified as:

Equation (2)

According to hypothesis 2, the battle loss rates of Party A and Party B are respectively

, 。

In this way, the mathematical model of regular operation is obtained:

Equation (3)

According to equation (3), the forces X (t) and Y (t) on both sides are monotonically decreasing functions, so it can be considered that the side whose force first decreases to zero is negative. In order to obtain the winning and losing conditions of both sides, it is not necessary to directly solve equation (3), but to discuss the changing law of phase trajectory on the phase plane, which can be obtained from equation (3).

(4)

The solution is

Ay2—bx2=k (5)

Note the initial conditions of equation (3). have

K=ay02—bx02 (6)

The phase trajectory determined by Equation (5) is a hyperbola, as shown in the figure. The arrow shows the changing trend of x(t) and y(t) with the increase of time t, so it can be seen that if k >；; 0, the trajectories will intersect on the y axis, that is, there is t 1 such that X (T 1) = 0 and Y (T 1) = > 0, that is, when Party A's strength is zero, Party B's strength is positive, indicating that Party B wins. Similarly, if K < 0, Party A wins, and if k=0, both sides draw.

Further analysis of the bid-winning conditions of one party, such as Party B, can be expressed as: (6) and indicate the meanings of A and B..

(7)

Equation (7) shows that the ratio y0/x0 of the initial forces of both sides affects the outcome of the war in a square relationship. For example, if Party B's strength is doubled (Party A remains unchanged), it will affect the original four times (px, RY, py remains unchanged). In order to compete with this, Party B only needs to double the original initial force y0. For this reason, the conventional war model is called the square rate model.

(1) For the first question. That is, in the conventional combat model, let the ratio of effective combat coefficients of Party A and Party B be a/b=4, and the initial forces x0 and y0 are the same. Ask Party B what the remaining strength is when winning, and how to determine the time for Party B to win. The solution is as follows:

According to the above stage trajectory:

The remaining strength of Party B when winning is: y(t)= 1

To determine the time for Party B to win t 1, it is necessary to solve Equation (3), which can be obtained as follows

Let x(t 1)=0. , you can calculate a/b=4.

T 1=, t1is directly proportional to Party A's operational effective derivative b.

The above is the answer to the first question, and the following is the answer to the second question:

(2) In the regular warfare model, let the ratio of the combat effectiveness coefficients of Party A and Party B be a/b=4, and the initial forces x0 and y0 are the same. If Party A has reserve troops to reinforce at a constant rate R after the battle starts, re-establish the model and discuss how to judge the outcome of both sides.

Solution: When Party A's reserve force increases at a constant rate R, the first equation of Equation (3) should be

That is, Equation (3) is changed to:

The phase trajectory is:

ay2—ry—bx2=k

k=ay02-ry-bx02

That is, the trajectory in the phase-trajectory diagram in the above figure moves upward by r/2a, from which it can be concluded that the equation condition for Party B to win is K >；; 0, that is:

Thinking and discussion:

In the war model, we applied the idea of differential equation modeling. We know that wars will always last for some time. With the development of the war situation, the manpower of the warring parties has also changed with time.

This model reflects the changes of the objects we describe over time. We take derivatives of variables with time to reflect their changing laws and predict their future forms. For example, in the war model, the first thing we need to describe is the change of the strength of both sides in unit time. We list differential equations by analyzing which factors are related to this change and the specific relationship between them. Then the relationship between the two sides is obtained by simplifying the equation. This is also the step of our differential equation modeling.

Is this all right?