Isoclines and Global Stability
129
The equilibrium plane [Eq. (5.7)] is parallel to the C axis and intersects the
P and Z axes at P = (1 + ulD)"IP L and Z = (I + oIDtU1p L • The intersection
between the equilibrium plane [Eq. (5.7)] and the surfaces defined byC= 0
and Z= 0 will generate isocline curves in the state space (Fig. 5.2).
The Algal Isocline. The important properties of the isocline curves in Fig.
5.2 are preserved by representing their projection on the (C, Z) plane. The
algal isocline, defined by setting C = 0 in Eq. (5.2), changes its shape with
increasing phosphorus loading (Fig. 5.3). At low P supply rates, the algal
isocline will be a strictly decreasing function. When the nutrient loading is
increased, the discontinuity caused by the piecewise linear functional
response [Eq. (5.5)] becomes more prominent. For algal biomass below the
incipient limiting food level (C < C1 the algal isocline will have a nonpositive
slope for all P loading levels; with increasing P loading, the slope of this part
of the isocline will increase and eventually become zero when the isocline
approaches the asymptotic zooplankton biomass level at the internal equilibrium {Z1. For C < C' the algal isocline develops a segment with positive
slope which becomes progressively larger with increasing P loading.
Rosenzweig (1969) has pointed out that such a "hump" on the prey isocline
will tend to destabilize the interaction between predator and prey.
The Grazer Isocline. Contrary to the algal isocline, the grazer isocline
maintains the same general shape with increasing P loading (Fig. 5.4). The
grazer isocline is a concave curve with a vertical left flank at the threshold
food level for positive population growth (C'1. As the total phosphorus
content of the system is finite, the P content of the algae will decrease as the
algal biomass increases. The decreasing food quality, in terms P content,
makes the grazer isocline bend down to the right, meeting the C axis at the
point where the algae become so nutritionally poor that they are unsuitable
for supporting grazer growth.
The shape of the grazer isocline in the present model is a direct consequence of considering phosphorus to be an essential and potentially limiting element for both algae and grazers, creating an interdependency
between the carrying capacities of predators and prey. In models without
such stoichiometric constraints on predator growth, the predator isocline
will simply either be a vertical straight line through the threshold food
concentration for positive predator growth, or a rectangular function
branching off to the right as the predator density reaches its carrying
capacity, as in Rosenzweig and Mac Arthur (1963).
Freedman and Wolkowicz (1986) have shown that the predator isocline
can have a negative slope at high prey densities if the prey has evolved
mechanisms of group defence. As an example of group defence they cite
that a lone musk ox can be successfully attacked by a pack of wolves, while
very few successful attacks are observed on larger herds of, say, six to eight
individuals. Freedman and Wolkowicz (1986) show that the presence of a
129
The equilibrium plane [Eq. (5.7)] is parallel to the C axis and intersects the
P and Z axes at P = (1 + ulD)"IP L and Z = (I + oIDtU1p L • The intersection
between the equilibrium plane [Eq. (5.7)] and the surfaces defined byC= 0
and Z= 0 will generate isocline curves in the state space (Fig. 5.2).
The Algal Isocline. The important properties of the isocline curves in Fig.
5.2 are preserved by representing their projection on the (C, Z) plane. The
algal isocline, defined by setting C = 0 in Eq. (5.2), changes its shape with
increasing phosphorus loading (Fig. 5.3). At low P supply rates, the algal
isocline will be a strictly decreasing function. When the nutrient loading is
increased, the discontinuity caused by the piecewise linear functional
response [Eq. (5.5)] becomes more prominent. For algal biomass below the
incipient limiting food level (C < C1 the algal isocline will have a nonpositive
slope for all P loading levels; with increasing P loading, the slope of this part
of the isocline will increase and eventually become zero when the isocline
approaches the asymptotic zooplankton biomass level at the internal equilibrium {Z1. For C < C' the algal isocline develops a segment with positive
slope which becomes progressively larger with increasing P loading.
Rosenzweig (1969) has pointed out that such a "hump" on the prey isocline
will tend to destabilize the interaction between predator and prey.
The Grazer Isocline. Contrary to the algal isocline, the grazer isocline
maintains the same general shape with increasing P loading (Fig. 5.4). The
grazer isocline is a concave curve with a vertical left flank at the threshold
food level for positive population growth (C'1. As the total phosphorus
content of the system is finite, the P content of the algae will decrease as the
algal biomass increases. The decreasing food quality, in terms P content,
makes the grazer isocline bend down to the right, meeting the C axis at the
point where the algae become so nutritionally poor that they are unsuitable
for supporting grazer growth.
The shape of the grazer isocline in the present model is a direct consequence of considering phosphorus to be an essential and potentially limiting element for both algae and grazers, creating an interdependency
between the carrying capacities of predators and prey. In models without
such stoichiometric constraints on predator growth, the predator isocline
will simply either be a vertical straight line through the threshold food
concentration for positive predator growth, or a rectangular function
branching off to the right as the predator density reaches its carrying
capacity, as in Rosenzweig and Mac Arthur (1963).
Freedman and Wolkowicz (1986) have shown that the predator isocline
can have a negative slope at high prey densities if the prey has evolved
mechanisms of group defence. As an example of group defence they cite
that a lone musk ox can be successfully attacked by a pack of wolves, while
very few successful attacks are observed on larger herds of, say, six to eight
individuals. Freedman and Wolkowicz (1986) show that the presence of a