A Holling’s type III response assumes that the consumer is limited by its abilit

Question

A Holling’s type III response assumes that the consumer is limited by its ability to process/handle food. It also assumes that at low food densities, consumption accelerates as food density increases. A Holling’s type III functional response can be modeled by the equation C(x)=ax^2/1+abx^2.

Show that C(x) is accelerating for low food densities and decelerating for large food densities. To do this, find a value x = a such that C''(a) = 0, C''(x) > 0 for 0 < x < a, and C''(x) < 0 for x > a.

Explanation / Answer

given C(x)=ax2/(1+ abx2)

differentiate with respect to x

C'(x)=[(2ax(1+ abx2))-(ax2(0+ 2abx))]/(1+ abx2)2

C'(x)=[2ax+ 2a2bx3-0- 2a2bx3]/(1+ abx2)2

C'(x)=2ax/(1+ abx2)2

differentiate with respect to x

C''(x)=[(2a(1+ abx2)2)-(2ax*2(1+ abx2)(0+2abx))]/(1+ abx2)4

C''(x)=[(2a(1+ abx2))-(2ax*2(0+2abx))]/(1+ abx2)3

C''(x)=[2a+ 2a2bx2-8a2bx2]/(1+ abx2)3

C''(x)=2a[1-3abx2]/(1+ abx2)3

C''(x)=0

2a[1-3abx2]/(1+ abx2)3=0

1-3abx2=0

x2=1/3ab

x=1/(3ab)

C''(1/(3ab)) = 0, C''(x) > 0 for 0 < x < 1/(3ab), and C''(x) < 0 for x > 1/(3ab).

so C(x) is accelerating for low food densities and decelerating for large food densities

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