Why must animals maintain a stable environment within their bodies?

Homeostasis, my friend.

Homeostasis is the process of maintaining a constant internal environment (temperature, water content, available energy, available oxygen & concentration of wastes in the blood.) Some animals are able to do this and are called endothermic or 'warm-blooded' (like humans). However, not all animals are able to do all of this - and they are referred to as ectothermic or 'cold-blooded.' 

Animals need to be able to maintain a stable internal environment because if they do not, they will most likely die. Animals need warmth to survive, so they need to be able to have a constant body temperature whether it be by using their body to create warmth or having to sit in the sun to keep warm,

Hope this helps. 

Mantle convection  is the slow  creepingmotion of earth's solid silicate  mantle  caused by  convection  currents carrying  heat from the interior  to the planet's surface.[1][2]  the earth's surface  lithosphere, which rides atop the  asthenosphere  (the two components of the  upper mantle), is divided into a number of  plates  that are continuously being created and consumed at their opposite  plate boundaries. accretion occurs as mantle is added to the growing edges of a plate, associated with  seafloor spreading. this hot added material cools down by  conduction  and  convection  of heat. at the consumption edges of the plate, the material has thermally contracted to become dense, and it sinks under its own weight in the process of  subduction  usually at an  ocean trench.[3]this subducted material sinks through the earth's interior. some subducted material appears to reach the lower mantle,[4]  while in other regions, this material is impeded from sinking further, possibly due to a phase transition from  spinel  to  silicate perovskiteand  magnesiowustite, an  endothermic reaction.[5]the subducted oceanic crust triggers  volcanism, although the basic mechanisms are varied. volcanism may occur due to processes that add buoyancy to partially melted mantle causing an upward flow due to a decrease in density of the partial melt.secondary forms of convection that may result in surface volcanism are postulated to occur as a consequence of intraplate extension[6]  and  mantle plumes.[7]it is because the mantle can convect that the  tectonic plates  are able to move around the earth's surface.[8]mantle convection seems to have been much more active during the  hadean  period, resulting in gravitational sorting of heavier molten  iron, and  nickel  elements and  sulphides  in the core, and lighter  silicate minerals  in the  mantle.