1. A MATHEMATICAL MODEL FOR A SELF-LIMITING POPULATION Glenn Ledder gledder@math.unl.edu University of Nebraska

2. THE CONCEPTUAL MODEL 1.A container holds p ( t ) microorganisms and a quantity w ( t ) of waste, with p (0) = p 0 « 1 and w (0) = 0. 2.Waste production is proportional to population. 3.In absence of waste, population growth is logistic. 4.Relative death rate increases linearly with waste amount.

3. THE MATHEMATICAL MODEL 2.Waste production is proportional to population. 3. In absence of waste, population growth is logistic. 4. Relative death rate increases linearly with waste amount. dw dt —– = kp w(0) = 0 p(0) = p 0 —– = rp ( 1– — ) – bwp dp dt pMpM

4. NONDIMENSIONALIZATION dw dt —– = kp w(0) = 0 p(0) = p 0 —– = rp ( 1– — ) – bwp dp dt pMpM Let p = MP w = — W τ = rt rbrb Then W′ = KP W(0) = 0 P′ = P(1–P–W) P(0) = P 0

5. NULLCLINES W is fixed when P = 0 and increasing when P > 0. P is fixed when P = 0 and P + W = 1. 1 1 P W

6. The DIFFERENTIAL EQUATION for the TRAJECTORIES Trajectories satisfy the equation —– = — = ——– dP dW 1-P-W K P′ W′ W′ = KPP′ = P(1–P–W) : We can combine the differential equations to obtain the differential equation for the trajectories in the phase plane. Givenand

7. CHANGE OF VARIABLES The trajectory equation, —– = ——–, is linear but not autonomous. dP dW 1-P-W K Let Z = P + W. Then K –— = 1 + K – Z. dZ dW This equation is autonomous as well as linear.

8. PHASE LINE Z = P + W.Z = P + W. K –— = 1 + K – Z dZ dW 1 + K1 + K Z 1 + K is a stable equilibrium solution; however, it is not achieved because W remains finite. P + W 0

9. TRAJECTORIES We can solve the trajectory equation to get P = 1 + K – W + (P0 – 1 – K) e-W/KP = 1 + K – W + (P0 – 1 – K) e-W/K K=0.5 K=1.0

10. MAXIMUM POPULATION P = 1 + K – W + (P0 – 1 – K) e-W/KP = 1 + K – W + (P0 – 1 – K) e-W/K The maximum population occurs when P + W = 1. P = 1 – W = 1 – K ln ——— 1+K-P 0 K

11. LIMITING BEHAVIOR With P 0 = 0 and P = 0, we have W = ( 1 + K )( 1 - e -W/K ). ( K, W ∞ ) = ( ——, —— ), where u = - ln ( 1 - s ) su - ssu - s s uu - ss uu - s

12. EXTINCTION TIME Let T be the time (in units of 1/ r ) at which P = 0. From W′ = KP, we have W∞/KW∞/K 0 T =T = du 1 + K – Ku + (P 0 -1-K) e -u —————————— P 0 =0.01