Math Insight

Let's examine the case where the competition is relatively weak, i.e., where $\alpha_{12}=\alpha_{21}=\frac{1}{2}$. To be concrete, we'll set $r_1=0.3$, $r_2=0.2$, $K_1=1500$, and $K_2=1200$ so that the competition model becomes \begin{align*} \diff{p_1}{t} & = 0.3 p_1 \left(1-\frac{p_1+\frac{1}{2}p_2}{ 1500 }\right)\\ \diff{p_2}{t} & = 0.2 p_2 \left(1-\frac{p_2+\frac{1}{2}p_1}{ 1200 }\right). \end{align*}

1. Calculate the nullclines. As before, each nullcline will be composed of two linear pieces.

   The $p_1$-nullcline:
   
   or
   

   The $p_2$-nullcline:
   
   or
   

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   Hint

   Enter $p_1$ as p_1 and $p_2$ as p_2.

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   (Although not necessary here, it will be easier to graph in the next step if you simplify the nullclines equations as much as possible.)

2. Graph the nullclines with the following phase plane applet. (Use the thick solid blue lines for the $p_1$-nullcline and the thin dashed green lines for the $p_2$-nullcline.)

   Feedback from applet
   Step 1: nullclines:
   Step 2: equilibria:
   Step 2: number of equilibria:
   Step 3: vector directions in regions:
   Step 3: vector locations in regions:
   Step 4: vector directions on nullclines:
   Step 4: vector locations on nullclines:
   Step 5: initial condition:
   Step 5: solution trajectory end point:
   Step 5: solution trajectory follows vector field:
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   Hint

   It might be easier to replace $p_2$ with $y$ and $p_1$ with $x$ in the equations for the nullclines. Then, you may be able to interpret the linear equations more readily.

   In terms of $x$ and $y$, the pieces of the $p_1$-nullcline are:
   
   or
   

   In terms of $x$ and $y$, the pieces of the $p_2$-nullcline are:
   
   or
   

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3. Graph the equilibria using the above phase plane applet. Set the step counter to 2, increase $n_e$ to reveal points for the equilibria, and move them to the correct locations. The equilibria are:
   
   
   (Enter the equilibria in any order, separated by commas, such as (1,3),(2,4),(5,6).)

4. Use the above phase plane applet to sketch direction vectors:

   1. in each region the phase plane separated by the nullclines (include only the biologically plausible quadratic of the phase plane), and
   2. and on each piece of each nullcline, as divided by the opposite nullcline (doing this separately for each line of the nullclines, include only the biologically plausible quadrant of the phase plane).

   Set the step to 3 to reveal vectors for each region of the phase plane. Move one vector to each region and set the corresponding general direction of each vector. Then, set the step counter to 4 to reveal vectors for each segment of the nullclines. Move one vector to each piece of the nullcline and set the corresponding general direction of each vector.

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   Hint

   One way to determine the direction vectors is to just test points $(p_1,p_2)$ in each region and nullcline segment, directly calculating $\diff{p_1}{t}$ and $\diff{p_2}{t}$ to determine the directions.

   Another way is to just reason about how the signs of $\diff{p_1}{t}$ and $\diff{p_2}{t}$ must change when crossing a nullcline. Here's one way to reason about the signs of $\diff{p_1}{t}$ and $\diff{p_2}{t}$ in the different regions. (All answer blanks in this section are optional.)

   At the $p_1$-nullcline, the sign of $\diff{p_1}{t}$ must be positive zero negative
   . Generally, as one crosses the $p_1$-nullcline, the sign of $\diff{p_1}{t}$ will stay the same switch
   . Similarly, for the the sign of $\diff{p_2}{t}$ when crossing the $p_2$-nullcline.

   We begin by deducing what the sign of $\diff{p_1}{t}$ must be on either side of the diagonal portion of the $p_1$-nullcline. (Since the vertical/horizontal lines of the nullclines border the biologically plausible quadrant of the nullclines, we don't have to worry about crossing those nullcline segments.)

   The one piece of $p_1$-nullcline that is a diagonal line is
   . In the region below this line, we know that $p_1 + \frac{1}{2}p_2 \lt$
   . Since we are in the biologically plausible quadrant, we know that $p_1 \gt$
   . Therefore, below the diagonal line of the $p_1$-nullcline, we can conclude that $\diff{p_1}{t}$ is zero negative positive
   . (Think about the signs of the factors of the expression for $\diff{p_1}{t}$.)

   This means that direction vectors below this diagonal line must point leftward rightward downward upward
   (although they may also point either up or down depending on the sign of $\diff{p_1}{t}$).

   On the other hand, above the diagonal line of the $p_1$-nullcline, the sign of $\diff{p_1}{t}$ must switch to being negative zero positive
   so the direction vectors switch to pointing rightward downward upward leftward
   .

   Here is similar reasoning for the sign of $\diff{p_2}{t}$ on either side of the diagonal line of the $p_2$-nullcline.

   The one piece of $p_2$-nullcline that is a diagonal line is
   . In the region below this line, we know that $p_2 + \frac{1}{2}p_1 \lt$
   . Since we are in the biologically plausible quadrant, we know that $p_2 \gt$
   . Therefore, below the diagonal line of the $p_2$-nullcline, we can conclude that $\diff{p_2}{t}$ is negative positive zero
   . This means that direction vectors below this diagonal line must point leftward upward rightward downward
   (although they may also point either left or right depending on the sign of $\diff{p_2}{t}$).

   On the other hand, above the diagonal line of the $p_2$-nullcline, the sign of $\diff{p_2}{t}$ must switch to being negative positive zero
   so the direction vectors switch to pointing leftward rightward upward downward
   .

   All four regions of the phase plane are defined as being above or below these two diagonal lines, so the above information is enough to deduce the direction of the vectors in each region.

   The same logic will work for determining the direction vectors on the nullcline. For example, for direction vectors on the $p_1$-nullcline, all you need to know is whether or not you are above or below the diagonal line of the $p_2$-nullcline. The above logic will then give you the sign of the $\diff{p_2}{t}$ in order to tell you if the direction vector should point straight left or right up or down
   

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5. Imagine that we start that species 2 is living alone and there are no individuals from species 1, i.e., that $p_1(0) =$
   . If we start with no individuals from species 1, what do we know must be true about $\diff{p_1}{t}$?
   $\diff{p_1}{t} =$
   Therefore, $p_1(t)$ must not be changing, and we can conclude that for all time, $p_1(t) =$
   .

   In this case, what happens to population 2? Since $p_1$ drops out from the equation, the dynamics for $p_2(t)$ becomes exactly the logistic equation with carrying capacity $K_2=$
   . Since we are assuming that we have some individuals in population 2, i.e., that $p_2(0) \gt 0$, what will happen to the population size of species 2 as time increases? $p_2(t)$ will approach
   .

6. Imagine that a small number of species 1 are introduced. One question is whether or not species 1 can invade the environment where species 2 is already established. Use the phase plane applet to sketch the solution for an initial condition where species 2 is well established and species 1 has just a small population. Use $p_2(0) = 700$ and $p_1(0)=50$.

   In the applet, change the step counter to 5, move the large green point to the initial condition and move the small cyan point so that the segment between them follows the general direction of the vector in the corresponding region. Increase nsegs to reveal most segments so that you continue to draw the solution trajectory $(p_1(t),p_2(t))$. When the solution hits the $p_2$-nullcline, it must be moving straight leftward upward downward rightward
   and then change directions (as it is in a region with a different representative direction vector). Think about if the solution can cross another nullcline and where it must go. End the trajectory at the point where the solution must be after a long time.

   Translate your results to plots of $p_1$ and $p_2$ versus time:

   Feedback from applet
   correct shapes:
   final values:
   initial conditions:

   Was species 1 able to successfully invade species 2? yes no
   

7. Now, imagine the reverse situation. Imagine that species 1 has been around for awhile and then species 2 attempts to invade. Use the phase plane to determine what a solution must look like for the initial condition of $p_1(0)=1300$ and $p_2(0)=50$.

   Since the above phase plane applet does not have the capability to draw two trajectories, you can either draw the trajectory $(p_1(t),p_2(t))$ on paper or imagine what it must look like. Then, translate your results to plots of $p_1(t)$ and $p_2(t)$ versus time:

   Feedback from applet
   correct shapes:
   final values:
   initial conditions:

   Was species 2 able to successfully invade species 1? yes no
   
   Did the final population sizes different depending on which species was present first? no yes
   

   These results are typical for the situation where the competition isn't too strong, i.e., the $\alpha_{12}$ and $\alpha_{21}$ aren't too large. In this case, the system favors as coexistence state, where both species persist and share the environment.

1. Calculate the nullclines.

   The $p_1$-nullcline:
   
   or
   

   The $p_2$-nullcline:
   
   or
   

2. Graph the nullclines with the following phase plane applet. (Use the thick solid blue lines for the $p_1$-nullcline and the thin dashed green lines for the $p_2$-nullcline.)

   Feedback from applet
   Step 1: nullclines:
   Step 2: equilibria:
   Step 2: number of equilibria:
   Step 3: vector directions in regions:
   Step 3: vector locations in regions:
   Step 4: vector directions on nullclines:
   Step 4: vector locations on nullclines:
   Step 5: initial condition:
   Step 5: solution trajectory end point:
   Step 5: solution trajectory follows vector field:
   Need help?Hide help

   Hint

   It might be easier to replace $p_2$ with $y$ and $p_1$ with $x$ in the equations for the nullclines. Then, you may be able to interpret the linear equations more readily.

   In terms of $x$ and $y$, the pieces of the $p_1$-nullcline are:
   
   or
   

   In terms of $x$ and $y$, the pieces of the $p_2$-nullcline are:
   
   or
   

   Hide help

3. Graph the equilibria using the above phase plane applet. Set the step counter to 2, increase $n_e$ to reveal points for the equilibria, and move them to the correct locations. The equilibria are:
   
   
   (Enter the equilibria in any order, separated by commas, such as (1,3),(2,4),(5,6).)

4. Use the above phase plane applet to sketch direction vectors:

   1. in each region the phase plane separated by the nullclines (include only the biologically plausible quadratic of the phase plane), and
   2. and on each piece of each nullcline, as divided by the opposite nullcline (doing this separately for each line of the nullclines, include only the biologically plausible quadrant of the phase plane).

   Set the step to 3 to reveal vectors for each region of the phase plane. Move one vector to each region and set the corresponding general direction of each vector. Then, set the step counter to 4 to reveal vectors for each segment of the nullclines. Move one vector to each piece of the nullcline and set the corresponding general direction of each vector.

5. We return to the question of whether or not species 1 can invade the environment where species 2 is already established. Use the phase plane applet to sketch the solution for an initial condition where species 2 is well established and species 1 has just a small population. Use $p_2(0) = 700$ and $p_1(0)=50$.

   In the applet, change the step counter to 5, move the large green point to the initial condition and move the small cyan point so that the segment between them follows the general direction of the vector in the corresponding region. Increase nsegs to reveal most segments so that you continue to draw the solution trajectory $(p_1(t),p_2(t))$. When the solution hits the $p_1$-nullcline, it must be moving straight rightward downward leftward upward
   and then change directions (as it is in a region with a different representative direction vector). Think about if the solution can cross another nullcline and where it must go. End the trajectory at the point where the solution must be after a long time.

   Translate your results to plots of $p_1$ and $p_2$ versus time:

   Feedback from applet
   correct shapes:
   final values:
   initial conditions:

   Was species 1 able to successfully invade species 2? no yes
   

6. Now, imagine the reverse situation. Imagine that species 1 has been around for awhile and then species 2 attempts to invade. Use the phase plane to determine what a solution must look like for the initial condition of $p_1(0)=1300$ and $p_2(0)=50$.

   Since the above phase plane applet does not have the capability to draw two trajectories, you can either draw the trajectory $(p_1(t),p_2(t))$ on paper or imagine what it must look like. Then, translate your results to plots of $p_1(t)$ and $p_2(t)$ versus time:

   Feedback from applet
   correct shapes:
   final values:
   initial conditions:

   Was species 2 able to successfully invade species 1? no yes
   
   Did the final population sizes different depending on which species was present first? no yes
   

   These results are typical for the situation where the competition is mutually strong, i.e., the $\alpha_{12}$ and $\alpha_{21}$ are large. In this case, either species could win. This condition can be referred to as founder control, as the winner tends to be the species that started with larger population size.

7. For this model, we've looked at easy initial conditions, i.e., conditions where one of the population sizes is small. But, what would happen if both population sizes are small or of similar size? In that case, the direction arrows that you've drawn in the phase plane don't give a clear answers. For example, if we started with 50 individuals of each species, $p_1(0)=50$ and $p_2(0)=50$, what would happen? Is it possible that both species could coexist together? Could the system approach the middle equilibrium?

   To answer this question, we need to learn how to analyze equilibria, in particular how to analyze the stability of equilibria. We need to develop more mathematical tools to understand the behavior of the system.

   In the mean time, execute the R script run_competition.R and then simulate the competition dynamical system with the run_competition command. (We assume that you already have the R package deSolve installed on your computer.) You can try different initial conditions. For example, to simulate with the initial condition $p_1(0)=50$ and $p_2(0)=700$, you can execute the command:
   run_competition(r1=0.3, r2=0.2, K1=2400, K2=3000, a12=2, a21=2, p10=50, p20=700)
   to confirm the results obtained above.

   If you simulate with the initial condition $p_1(0)=p_2(0)=50$, which population wins? both species die out species 1 species 2 the species coexist
   If you simulate with the initial condition $p_1(0)=p_2(0)=150$, who wins the competition? the species coexist both species die out species 2 species 1
   Can you find an initial condition (with the strong competition parameters we have here) where the species coexist, i.e., where neither population size tends to zero as $t$ increases?