MBriereE function

Modified Briere Equation

MBriereE is used to calculate $y$ values at given $x$ values using the modified Brière equation or one of its simplified versions. UTF-8

MBriereE(P, x, simpver = 1)

Arguments

• P: the parameters of the modified Brière equation or one of its simplified versions.
• x: the given $x$ values.
• simpver: an optional argument to use the simplified version of the modified Brière equation.

Details

When simpver = NULL, the modified Brière equation is selected:

$$\mbox{if } x \in{\left(x_{\mathrm{min}}, \ x_{\mathrm{max}}\right)},$$ $$y = a\left|x(x-x_{\mathrm{min}})(x_{\mathrm{max}}-x)^{1/m}\right|^{\delta};$$ $$\mbox{if } x \notin{\left(x_{\mathrm{min}}, \ x_{\mathrm{max}}\right)},$$ $$y = 0.$$

Here, $x$ and $y$ represent the independent and dependent variables, respectively; and $a$, $m$, $x_{\mathrm{min}}$, $x_{\mathrm{max}}$, and $\delta$ are constants to be estimated, where $x_{\mathrm{min}}$ and $x_{\mathrm{max}}$ represents the lower and upper intersections between the curve and the $x$-axis. $y$ is defined as 0 when $x < x_{\mathrm{min}}$ or $x > x_{\mathrm{max}}$. There are five elements in P, representing the values of $a$, $m$, $x_{\mathrm{min}}$, $x_{\mathrm{max}}$, and $\delta$, respectively.

$\quad$ When simpver = 1, the simplified version 1 is selected:

$$\mbox{if } x \in{\left(0, \ x_{\mathrm{max}}\right)},$$ $$y = a\left|x^{2}(x_{\mathrm{max}}-x)^{1/m}\right|^{\delta};$$ $$\mbox{if } x \notin{\left(0, \ x_{\mathrm{max}}\right)},$$ $$y = 0.$$

There are four elements in P, representing the values of $a$, $m$, $x_{\mathrm{max}}$, and $\delta$, respectively.

$\quad$ When simpver = 2, the simplified version 2 is selected:

$$\mbox{if } x \in{\left(x_{\mathrm{min}}, \ x_{\mathrm{max}}\right)},$$ $$y = ax(x-x_{\mathrm{min}})(x_{\mathrm{max}}-x)^{1/m};$$ $$\mbox{if } x \notin{\left(x_{\mathrm{min}}, \ x_{\mathrm{max}}\right)},$$ $$y = 0.$$

There are four elements in P representing the values of $a$, $m$, $x_{\mathrm{min}}$, and $x_{\mathrm{max}}$, respectively.

$\quad$ When simpver = 3, the simplified version 3 is selected:

$$\mbox{if } x \in{\left(0, \ x_{\mathrm{max}}\right)},$$ $$y = ax^{2}(x_{\mathrm{max}}-x)^{1/m};$$ $$\mbox{if } x \notin{\left(0, \ x_{\mathrm{max}}\right)},$$ $$y = 0.$$

There are three elements in P representing the values of $a$, $m$, and $x_{\mathrm{max}}$, respectively.

Returns

The $y$ values predicted by the modified Brière equation or one of its simplified versions.

Note

We have added a parameter $\delta$ in the original Brière equation (i.e., simpver = 2) to increase the flexibility for data fitting.

References

Brière, J.-F., Pracros, P, Le Roux, A.-Y., Pierre, J.-S. (1999) A novel rate model of temperature-dependent development for arthropods. Environmental Entomology 28, 22$-$29. tools:::Rd_expr_doi("10.1093/ee/28.1.22")

Cao, L., Shi, P., Li, L., Chen, G. (2019) A new flexible sigmoidal growth model. Symmetry 11, 204. tools:::Rd_expr_doi("10.3390/sym11020204")

Jin, J., Quinn, B.K., Shi, P. (2022) The modified Brière equation and its applications. Plants 11, 1769. tools:::Rd_expr_doi("10.3390/plants11131769")

Shi, P., Gielis, J., Quinn, B.K., Niklas, K.J., Ratkowsky, D.A., Schrader, J., Ruan, H., Wang, L., Niinemets, Ü. (2022) 'biogeom': An R package for simulating and fitting natural shapes. Annals of the New York Academy of Sciences 1516, 123$-$134. tools:::Rd_expr_doi("10.1111/nyas.14862")

Author(s)

Peijian Shi pjshi@njfu.edu.cn , Johan Gielis johan.gielis@uantwerpen.be , Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca .

See Also

areaovate, curveovate, fitovate, fitsigmoid, MbetaE, MLRFE, MPerformanceE, sigmoid

Examples

x2   <- seq(-5, 15, len=2000)
Par2 <- c(0.01, 3, 0, 10, 1)
y2   <- MBriereE(P=Par2, x=x2, simpver=NULL)

dev.new()
plot( x2, y2, cex.lab=1.5, cex.axis=1.5, type="l",
      xlab=expression(italic(x)), ylab=expression(italic(y)) )

graphics.off()