At equilibrium when ds / dt = 0 and dn / dt = 0:
set equations (9) and (10) to zero, and s = s1 c = c1:
| DERIVATION ASSISTANCE | ||
| equation | notes | text equation |
At equilibrium when ds / dt = 0 and dn / dt = 0: |
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(8C) | |
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(8D) | |
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(8E) | |
| To derive text equations 11 and 12, the plant growth and herbivore population isoclines, two plant model; set equations (9) and (10) to zero, and s = s1 c = c1: |
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[from (9)] | (11B) |
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(11C) | |
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(11D) | |
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[equals equation (11)] | (11E) |
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[from (10)] | (12A) |
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(12B) | |
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(12C) | |
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(12D) | |
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[equals equation 12] | (12E) |
| To derive equation 16, the plant growth isocine for logistic plant growth; two plant model: | ||
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(16A) | |
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(16B) | |
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(16C) | |
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(16D) | |
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(16E) | |
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(16F) | |
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[equals equation 16] | (16G) |
| To derive equation 19, equilibrium plant size, multiple plant model. | ||
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[from equations 6 and 8, multiple plant model; D(i) = s / s + s0] | (19A) |
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[add logistic plant growth] | (19B) |
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(19C) | |
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dsi / dt = 0 at equilibrium | (19D) |
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(19E) | |
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[equals equation 19] | (19F) |
| Derive equation 20, herbivore production by plant i, logistic plant growth model. | ||
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[equation 19; cm = ci at si = 0; definition of cm] | (20A) |
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(20B) | |
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[definition of P(i); si/S cancels out p] | (20C) |
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[substitute equation (19)] | (20D) |
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[substitute equation 20B] | (20E) |
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[equals equation 20] | (20F) |
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