3 #1. x' = x(1 - x - y) Check if the two species co-exist or not. 3 #2. x = x y ==+ X 2 2 Sketch the phase portrait for the predator-prey system.
Answers
1. To check if the two species co-exist in the system x' = x(1 - x - y), we need to find the equilibrium points where both species' populations remain constant. The equilibrium points occur when x'=0 and y'=0:
x'(x,y) = x(1 - x - y) = 0
To solve for the equilibrium points, we can examine the possible scenarios:
- x = 0: This means that there are no individuals of species x, and thus the two species do not co-exist.
- 1 - x - y = 0: This equation represents the situation where the two species co-exist.
2. For the predator-prey system described by x' = xy and y' = -y + x^2/2, we need to sketch the phase portrait to visualize the system's behavior. A phase portrait is a plot of the trajectories in the phase plane (x,y), which represent the populations of the two species over time.
To draw the phase portrait, follow these steps:
a. Find the equilibrium points where x' = 0 and y' = 0:
x' = xy = 0 and y' = -y + x^2/2 = 0
b. Analyze the stability of the equilibrium points.
c. Sketch the trajectories by examining the direction of the vectors (x', y') at various points in the phase plane.
For question 1, we can use the concept of equilibrium points to determine if the two species co-exist or not. An equilibrium point is a point in the phase space where the population of both species does not change over time. To find the equilibrium points, we set both derivatives equal to zero:
x' = x(1 - x - y) = 0
y' = y(-2 + x + y) = 0
From the first equation, we have three possible equilibrium points: (0,0), (1,0), and (x, y) = (1-x, 0). From the second equation, we have two possible equilibrium points: (0,0) and (2,0).
To determine if the two species co-exist or not, we need to look at the stability of these equilibrium points. A stable equilibrium point means that if the populations are close to that point, they will stay close to that point over time. An unstable equilibrium point means that if the populations are close to that point, they will move away from that point over time.
For the (0,0) equilibrium point, both derivatives are zero, which means that both populations are at zero. This is an unstable equilibrium point because any slight perturbation will cause the populations to either increase or decrease.
For the (1,0) equilibrium point, x' is zero and y' is negative, which means that the x population will stay constant and the y population will decrease. This is a stable equilibrium point for the x species, but an unstable equilibrium point for the y species.
For the (1-x, 0) equilibrium point, x' is zero and y' is negative, which means that the x population will increase and the y population will decrease. This is a stable equilibrium point for both species.
For the (2,0) equilibrium point, x' is positive and y' is zero, which means that the x population will increase and the y population will stay constant. This is an unstable equilibrium point for the y species.
Based on the stability analysis, we can conclude that the two species can co-exist at the (1-x, 0) equilibrium point, but not at any of the other equilibrium points.
For question 2, we have the predator-prey system x' = x - xy and y' = xy - y^2. To sketch the phase portrait, we first find the equilibrium points by setting both derivatives equal to zero:
x' = x - xy = 0
y' = xy - y^2 = 0
From the first equation, we have two possible equilibrium points: (0,0) and (1,1). From the second equation, we have two possible equilibrium points: (0,0) and (y, x-y) where y is any positive number.
To determine the stability of the equilibrium points, we can use the Jacobian matrix:
J = [1-y -x]
[y x-2y]
At the (0,0) equilibrium point, J = [1 0; 0 0], which has eigenvalues 1 and 0. This is a saddle point, which means that the trajectories of the system approach the equilibrium point in some directions and move away in other directions.
At the (1,1) equilibrium point, J = [0 -1; 1 -1], which has eigenvalues -1 and 0. This is a stable node, which means that the trajectories of the system approach the equilibrium point and stay close to it over time.
At the (y, x-y) equilibrium point, J = [1-y -x+y; y -2y], which has eigenvalues (-1 ± sqrt(1+4y))/2. If y > 0, then both eigenvalues are negative, which means that the equilibrium point is a stable node. If y < 0, then one eigenvalue is positive and one eigenvalue is negative, which means that the equilibrium point is a saddle point.
Based on the stability analysis, we can sketch the phase portrait as shown in the attached image. The trajectories of the system approach the (1,1) equilibrium point and stay close to it over time.
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Related Questions
Kevin runs a café. Every day the café is open he earns money in sales and spends money on supplies. After costs, how much more money did Kevin make on Saturday than on Friday?
Day Sales:
Monday $512.87
Friday $735.90
Saturday $807.31
Supply Costs
$200.92
$232.86
$289.00
Answers
Answer:
$15.27
Step-by-step explanation:
You want to know how much more profit Kevin made on Saturday than on Friday, if his revenue and expenses for the two days were ...
Saturday: $807.31 revenue; 289.00 expensesFriday: $735.90 revenue; $232.86 expensesProfitThe profit each day is the difference between revenue and expenses:
Saturday profit = $807.31 -289.00 = $518.31
Friday profit = $735.90 -232.86 = $503.04
DifferenceThe profit on Saturday exceeded the profit on Friday by ...
$518.31 -503.04 = $15.27
Kevin made $15.27 more on Saturday than Friday.
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6. What is the perimeter of the trapezoid if x = 9 and y = -3? Round to the nearest whole number.
x2 + 3y
3(x + Ey)2
3
(x + y)?
X-5y
-2y
310
292
Ο Ο Ο Ο
940
289
Answers
In a recent survey of college students, it was determined that 4 out of 5 students drink coffee. Of those students who drink coffee, 1 out of 8 students adds cream. If a college campus has 18000 students, how many students drink coffee with cream?
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The number of students who drink coffee with cream is 1800.
What is an expression?Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.
Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols, which can also be used to indicate the logical syntax's order of operations and other features.
Given that in a recent survey of college students, it was determined that 4 out of 5 students drink coffee. Of those students who drink coffee, 1 out of 8 students adds cream. If a college campus has 18000 students.
The number of students who drink coffee with cream is calculated as:-
Number = ( 4 / 5 ) x ( 1 / 8 ) x 18000
Number = 1800
Therefore, there are 1800 students who like their coffee with cream.
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Help please ...A square field has an area of 479 ft2. What is the approximate length of a side of the field? Give your answer to the nearest foot. Explain your response.
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you do 479 x 2 = 958 there is your answer your welcome
need answer shortly pleaaaaaaaaaase
Answers
10.041 , 10.1 , 10.81 , 10.857 , 11.022
Given the function f(x)= e^3x, write an expression that represents the derivative of f using the limit shown below.
Answers
Answer:
f'(x)=6x+8
Step-by-step explanation:
Look at the arithmetic sequence below. Use the boxes to create a recursive formula for the sequence. 18, 21, 24, 27, 30, 33... Response area =
Answers
Answer:
we have:
u(1) = 18
u(2) = 21 = 18 + 3
u(3) = 24 = 18 + 3.2
u(4) = 27 = 18 + 3.3
.......
=> the sequence has: u(1) = 18
d = 3
=> u(n) = 18 + 3(n - 1) = 3n + 15
so the recursive formula for the sequence is u(n) = 3n + 15
Step-by-step explanation:
there are about 1,265 school districts in the state of texas that generate an equivalent expression using prime factorization for the number 1,265
pls, help whoever answers first gets brainliest!!
Answers
Answer:
1265 = 5*11*23------------------------------------------
Find prime factors of 1265.
We see it is divisible by 5 since ends with 5:
1265/5 = 253253 is not divisible by 3 or 9 since the sum of digits is not divisible by 3 or 9. It is not divisible by 7 (since 25 - 3*2 = 19 is not divisible by 7) but divisible by 11, since the sum of digits in odd places is same as the digit in the even place (recall divisibility by 11):
253/11 = 2323 is a prime number so no more factors.
Hence the prime factorization of 1265 is:
1265 = 5*11*23What is the average rate of change of f(x) = −x2 + 3x + 6 over the interval −3 ≤ x ≤ 3? A. −2 B. −1 C. 3 D. 6
Answers
A function is a relation between a set of inputs and a set of possible outputs, where each input is uniquely associated with a single output.the average rate of change of f(x) over the interval \([-3, 3]\) is:
\((6 - (-12)) / 6 = 18/6 = 3\) Thus, option C is correct.
What is the average rate of change of f(x)?The average rate of change of a function over an interval is given by the difference in the function values at the endpoints of the interval, divided by the length of the interval.
Therefore, to find the average rate of change of \(f(x) = − + 3x + 6\) over the interval \(−3 ≤ x ≤ 3\), we need to evaluate the function at the endpoints of the interval and then divide by the length of the interval.
\(f(-3) = + 3(-3) + 6 = -9 - 9 + 6 = -12\)
\(f(3) = + 3(3) + 6 = -9 + 9 + 6 = 6\)
The length of the interval is 3 - (-3) = 6.
Therefore, the average rate of change of f(x) over the interval \([-3, 3]\) is:
\((6 - (-12)) / 6 = 18/6 = 3\)
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PLEASE I NEED HELP RN
use the elimination method to solve. -7x+y=-19 -2x+3y=-19
Answers
Answer:
(x, y) = (2, -5)
Step-by-step explanation:
The coefficient of y in the second equation is an integer multiple of the coefficient in the first equation, so it is convenient to eliminate y. Subtracting 3 times the first equation from the second, we have ...
(-2x +3y) -3(-7x +y) = (-19) -3(-19)
-2x +3y +21x -3y = -19 +57 . . . . . . . eliminate parentheses
19x = 38 . . . . . . . . . simplify
x = 2 . . . . . . . divide by 19
-7(2) +y = -19 . . . . . substitute for x in the first equation
y = -5 . . . . . . add 14
The solution is (x, y) = (2, -5).
An engineer wrote down the following estimates in his log book: 120, 130, 150, 180, 190 The median of this data set is 150. If two numbers less that 120 were added to the data set, how would that affect the median? *
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Answer: The median would be 130.
Given the equation f + 24 = −3, solve for f.
Answers
Answer:
f=-27, hope this helped my love have a good rest of your day ^^
Step-by-step explanation:
subtract 24 from both sides
the you get f=-27
Write the equation of a line through point (-5,1) and perpendicular to 2x - 3y = 6.
Answers
Answer:
y = 2/3x - 2
Step-by-step explanation:
Step 1:
y = mx + b Slope Intercept Form
Step 2:
2x - 3y = 6 Equation
Step 3:
- 3y = - 2x + 6 Subtract 2x on both sides
Step 4:
y = 2/3x - 2 Divide - 3 on both sides
Answer:
y = 2/3x - 2
Hope This Helps :)
ok, im failing math rn so plz help
Answers
Answer:
-3/4
Step-by-step explanation:
Point A is at (-4,3) and Point B is at (4,-3)
The slope is at
m = (y2-y1)/(x2-x1)
= (-3 -3)/(4 - -4)
= (-3-3)/(4+4)
= -6/8
= -3/4
Each day three church bells are rung in a random order. what is the probability that the smallest bell rings first three days in a row?
Answers
The probability that the smallest bell rings the first three days in a row is 1/27. when Each day three church bells are rung in a random order
In the given data there are 3 bells in the church in which there is a small bell and the three bells are rung in a random order. we need to find the probability that the smallest bell rings the first three days in a row.
The probability that the smallest bell rings on any given first day can be given as = 1/3
Because there are three bells and each bell has an equal chance of being rung first. The probability that this happens three days in a row is given as
= (1/3) × (1/3) × (1/3)
= 1/27
Therefore, the probability that the smallest bell rings the first three days in a row is 1/27
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Please help me with these questions. Thank you!
Answers
Answer:
382.19cm²
Step-by-step explanation:
a / sin(A) = b / sin(B)
a / sin(68) = 53 / sin(95)
a = (53 x sin(68)) / sin(95)
a = 49.33
area = ½absin(C)
C = 180 - (95 + 68)
C = 180 - 163
C = 17°
area = ½(53)(49.33)sin(17)
area = 382.19cm²
the map ratio of a map is 1: 50000. find the length on the ground represented by 6.4cm on the map
Answers
Answer:
320,000
Step-by-step explanation:
6.4cm on the map would be 320,000 (ft I'm assuming) and I found that by multiplying 50,000x6.4.
Preston bought 16 donuts. There were d donuts in each box. Choose the expression that shows how many boxes of donuts Preston bought
Answers
Answer:
What are the option?
Step-by-step explanation:
please tell me the answer
Answers
A Dutch company built a submarine that can hold 10 passengers and 1 pilot. The submarine has a safety feature that will automatically raise the submarine if it goes below the certified depth of −650feet. To show this feature to a buyer the submarine dove to a depth of −872\(\frac{5}{12}\)feet. The automatic feature then raised the submarine 50\(\frac{3}{5}\)feet per minute. How much time, in minutes, does it take for the submarine to reach the certified depth? Round your answer to the nearest tenth of a minute.
Answers
The time, in minutes that it take for the submarine to reach the certified depth is 12.85 minutes.
How to illustrate the information?From the information, the submarine has a safety feature that will automatically raise the submarine if it goes below the certified depth of −650feet.
It should be noted that the automatic feature then raised the submarine 50 3/5 feet per minute.
Therefore, the time needed to reach the depth will be:
= 650 / 50 3/5
= 650 / 50.6
= 12.85 minutes
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Hi can i get some help with these pleasee? thank you!!
Write an inequality to represent the situation.
Solve the inequality to answer the question.
36) Jan is renting one of the VeoRide bikes in downtown Nashua. The bikes cost $0.50 for each quarter of an hour. Jan has less than $5 for the ride. Let q represent the number of quarter hours Jan rides. How many can she ride and still be under budget?
37) Toby is trying to keep at least $100 in his piggy bank. At the start of the summer he had $436. Each week he takes $12 out of his piggy bank for expenses. Let w represents the number of weeks. How many weeks can Toby keep up this habit?
Answers
Answer:
Step-by-step explanation:
36) 0.50q < 5
0.50/0.50 q < 5/0.50
q < 10
37) 436 - 12w ≥ 100
-12w ≥ 100 - 436
-12w ≥ -336
-12 / -12 w ≤ -336/ -12
w ≤ 28
I Need Help With This Question
Answers
Answer:
Step-by-step explanation:
Dont do it. Just take the detention
What are the domain and the range of this graph?
Answers
The Domain is 1≤ x ≤ 5 and range is 1≤ y ≤ 5.
What is domain and range of the graph?The sets of all the x-coordinates and all the y-coordinates of ordered pairs, respectively, are the domain and range of a relation.
A function's range is its potential output, whereas its domain is the set of all possible input values.
Given:
As, domain is the input values shown on the x-axis
Then the x- axis values starts from x=1 to x=5.
So, Domain= 1≤ x ≤ 5
and, the range is the output values, which are shown on the y-axis.
Then, the y- axis values ranges from y=1 to y=5.
So, Range = 1≤ y ≤ 5
Hence, the Domain is 1 ≤ x ≤ 5 and range is 1 ≤ y ≤ 5
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The total grams of dietary fiber, f , in p apples can be represented by the equation f=4.4p . What is the constant of proportionality of the equation? Responses
Answers
The equation relating the total grams of dietary fiber (f) to the number of apples (p) is:
f = 4.4p
The constant of proportionality in this equation is the coefficient of p, which is 4.4. Therefore, the constant of proportionality is 4.4.
Solve the given differential equation. (6x+1) y^2 dy/dx + 4x^2 + 2y^3 = 0.
Answers
To solve the given differential equation (6x+1)y^2 dy/dx + 4x^2 + 2y^3 = 0, we can rearrange the terms to make it an explicit equation for dy/dx: dy/dx = -(4x^2 + 2y^3) / ((6x+1)y^2)
Now, separate the variables by moving all the x terms to one side and y terms to the other side: (dy / (y^2 - 2y^3)) = - (4x dx / (6x + 1))
Next, integrate both sides of the equation: (dy / (y^2 - 2y^3)) = -∫(4x dx / (6x + 1))
To solve the given differential equation (6x+1) y^2 dy/dx + 4x^2 + 2y^3 = 0, we can rearrange the terms to get:
(6x+1) y^2 dy = - (4x^2 + 2y^3) dx
Now, we can integrate both sides:
∫ (6x+1) y^2 dy = - ∫ (4x^2 + 2y^3) dx
Integrating the left-hand side with respect to y and the right-hand side with respect to x, we get:
2y^3 (3x + 1) = - (4/3)x^3 - y^4 + C
where C is the constant of integration.
To solve for y, we can isolate y^4 on one side of the equation and take the fourth root:
y^4 = (4/3)x^3 - 2y^3 (3x + 1) + C
y^4 + 6y^3 x + (4/3)x^3 - C = 0
This is a quartic equation in y^4, which can be difficult to solve. However, if we substitute z = y^3, we can rewrite the equation as:
z^2 + 6xz + (4/3)x^3 - C = 0
This is a quadratic equation in z, which can be solved using the quadratic formula:
z = (-6x ± sqrt(36x^2 - 4(4/3)x^3 + 4C)) / 2
Simplifying and substituting back for y, we get:
y = (z)^(1/3)
y = [(-6x ± sqrt(36x^2 - 4(4/3)x^3 + 4C)) / 2]^(1/3)
This is the general solution to the given differential equation. To find a particular solution, we need to know the initial condition, such as y(0) = 1.
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7. Translate up two units, and left 4 units.c7Original NewCoordinates: Coordinates:A:(2) A: ()B: (___) B:(___)C: (_) C:(___)D:(_) D': (__)E: () E':(5BD3A35los.
Answers
Consider the if a point is moved up, then its y-coordinate increases. While if the point is moved left then the x-coordinate decreases.
So the transformation corresponding to the movement of a point up by 2 units and left by 4 units can be described as follows,
\((x,y)\rightarrow(x-4,y+2)\)The original coordinates (x,y) of points can be obtained from the graph, and the transformation can be applied to obtain the new coordinates.
The coordinates of point A will be as follows,
\(\begin{gathered} A\colon(4,1) \\ A^{\prime}\colon(0,3) \end{gathered}\)The coordinates of point B will be,
\(\begin{gathered} B\colon(8,5) \\ B^{\prime}\colon(4,7) \end{gathered}\)The coordinates of point C will be,
\(\begin{gathered} C\colon(4,7) \\ C^{\prime}\colon(0,9) \end{gathered}\)The coordinates of point D will be,
\(\begin{gathered} D\colon(5,4) \\ D^{\prime}\colon(1,6) \end{gathered}\)The coordinates of point E will be,
\(\begin{gathered} E\colon(2,3) \\ E^{\prime}\colon(-2,5) \end{gathered}\)Thus, the original and new coordinates of each point are obtained.
Evaluate 64^1/2x
10^-2
Give your answer as a fraction in its simplest form.
Answers
\((64)^{\tfrac 12} \times 10^{-2}=(8^2)^{\tfrac 12} \times \dfrac 1{10^2}=\dfrac{8}{100} = \dfrac{2}{25}\)
the mean price is 520000 and the stnadard deviation is 58000. at least what percent of homes would you expect to be priced between 418500 and 621500?
Answers
It can be stated that a minimum of 90.82% of houses would fall within the price range of $418,500 to $621,500.
The given data are as follows:
Mean price (μ) = $520000
Standard deviation (σ) = $58000
Price range: $418500 to $621500
We are to find the percentage of homes priced between $418500 and $621500.To find the required percentage, we first need to standardize the given range of prices by converting them into z-scores.
The z-score formula is z = (x - μ) / σwhere x is the price, μ is the mean price, and σ is the standard deviation. So, for the lower limit: z₁ = (418500 - 520000) / 58000 = -1.75 And for the upper limit: z₂ = (621500 - 520000) / 58000 = 1.75
Now, we need to find the area under the normal curve between these two z-scores, which represents the percentage of homes priced between $418500 and $621500. To do this, we can use a calculator.
The area between $418500 and $621500 corresponds to the area between z₁ and z₂ on the standard normal distribution curve. The area between z₁ and z₂ is 0.9082 (rounded to 4 decimal places).
Therefore, we can say that at least 90.82% of homes would be priced between $418500 and $621500.
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one box contains 4 green balls and 3 red balls. another box contains 4 redballs and 3 purple balls. a box will be selected at random, then two balls willbe randomly selected from that box without replacement. if the first ball isred, what is the probability that the second ball also will be red? express youranswer as a common fraction and sum the numerator and denominator.
Answers
The probability of drawing two red balls in a row from one of the two boxes, given that the first ball drawn is red, is 1/7.
The first step in solving this problem is to determine the probability of selecting the red box versus the other box. Since we have two boxes and each one has an equal chance of being selected, the probability of selecting the red box is 1/2.
Now we can put all of these probabilities together to determine the overall probability of drawing two red balls in a row from one of the two boxes.
We know that the probability of selecting the red box is 1/2, the probability of drawing a red ball on the first try is 4/7, and the probability of drawing a red ball on the second try, given that we've already drawn one red ball, is 1/2.
To find the overall probability, we simply multiply these three probabilities together:
(1/2) x (4/7) x (1/2) = 1/7
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The probability that the second ball will be red, given that the first ball is red, is 3/7.
Explanation:To find the probability that the second ball will also be red given that the first ball is red, we need to consider two cases.
Case 1: Selecting a box with 4 red balls and 3 purple balls.
The probability of selecting a red ball from this box is 4/7. Since we are drawing without replacement, the probability of selecting another red ball is 3/6.Case 2: Selecting a box with 4 green balls and 3 red balls.
The probability of selecting a red ball from this box is 3/7. Since we are drawing without replacement, the probability of selecting another red ball is 2/6.Now, we calculate the overall probability by adding the probabilities of the two cases: (4/7) * (3/6) + (3/7) * (2/6) = 12/42 + 6/42 = 18/42 = 3/7.
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Need help on this asap pleaseee
Answers
The rule each sister wrote should be matched with the diagram that represent the rule as follows;
Ella ⇒ p = 2t + 2.
Della ⇒ p = 2(t + 1)
Stella ⇒ p = 2(t - 1) + 4
What is a linear function?In Mathematics and Geometry, a linear function refers to a type of function whose equation is graphically represented by a straight line on the cartesian coordinate (graph).
Assuming the variable t represent the number of smaller tables pushed together and the variable p represent the number of people that can be seated, a linear function (rule) that models the thinking of each sister can be written as follows;
Ella ⇒ p = 2t + 2.
When t = 1, the value of p is given by;
p = 2(1) + 2 = 4.
When t = 2, the value of p is given by;
p = 2(2) + 2 = 6.
When t = 3, the value of p is given by;
p = 2(3) + 2 = 8.
For Della, we have:
p = 2(t + 1)
When t = 1, the value of p is given by;
p = 2(1 + 1) = 4.
When t = 2, the value of p is given by;
p = 2(2 + 1) = 6.
When t = 3, the value of p is given by;
p = 2(3 + 1) = 8.
For Stella, we have:
p = 2(t - 1) + 4
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