Figure 1.
Killing-the-Winner (KtW) model with and without partial defense.
Original KtW model with complete defense (no predation on defense specialist, A) and modified version with partial defense analyzed here (B). The mortality rate of the predator or parasite is indicated with a horizontal arrow. The total nutrient content in the system is the sum of N, C, D and P.
Table 1.
Mass balance equations and equilibrium solutions for competition specialist (C), defense strategist (D), predator (P) and free nutrients (N) for original and modified KtW with partial defense.
Table 2.
Symbols and parameter values used including trade-off functions for defensive and competitive abilities of the defense strategist.
Figure 2.
Trade-off functions between competitive and defensive abilities of the defense strategist.
Relative affinity of defense strategist and clearance rate of predator on the defense strategist with respect to the defense strategy (
0 pure competition,
1 pure defense). For a trade-off parameter
of 1 (dashed line), a linear trade-off shape is obtained where the loss in competitive ability (i.e. reduction of affinity of the defense strategist) is proportional to the gain in defense (i.e. the reduction of the predator's clearance rate) as the strategy
increases. For a trade-off parameter
below 1 (solid lines, shown for
), a trade-off is obtained where the clearance rate drops initially more steeply than the affinity for increasing
, illustrating that a lot is gained initially in terms of reduced predation for a small reduction in competitive ability. The extension to a high trade-off parameters (
) is trivial (i.e. the initial gain in defense is small relative to the loss in competition), but not of interest here since solutions with the defense strategist present only exist for
(not shown).
Figure 3.
Biomass distributions at steady state as a function of defense strategy and trade-off parameter
.
Steady-state biomass distributions for the predator (P*, top), the defense strategist (D*, middle) and the competition specialist (C*, bottom) with respect to the defense strategy and trade-off parameter
for three limiting nutrient contents (
20, left,
50, middle, and
80, right). Other parameters as in Table 2.
Figure 4.
Biomass sections as a function of the defense strategy for given trade-off parameters
.
Steady-state biomass of competition specialist (C*, fine dotted lines), defense strategist (D*, dashed line) and predator (P*, solid line) as a function of defense strategy for different trade-offs ( top,
middle, and
bottom) for
Other parameters as in Table 2.
Figure 5.
Optimal defense strategies with respect to maximum biomass, maximum production and evolutionarily stable strategy (ESS).
Defense strategies corresponding to defense strategist's maximum biomass (blue), maximum production (defined as
, green) and ESS (red) are shown as a function of the trade-off parameter
for different nutrient contents. The ESS is defined by the maximum net growth rate of a invading mutant, which is found by critical point analysis of the first partial derivative of the net growth rate with respect to strategy
(see Appendix S1). Different contours show the effect of the total nutrient content
on the maximizing strategies. Other parameters as in Table 2.